Clothing can also serves as protection. For example, I do not wear shoes out of modesty, I use them because I don't want punctures and scrapes and thorns in my feet. Cowboys wore boots, chaps, hats and gloves to protect themselves, against snake bite, hand damage when handling rough rope with strong herd animals, and shade for sunstroke. Head covering can also retain body heat in the cold.
Of course the first function of loincloths is to protect sensitive genitalia, not hide it. The hiding happens as a result of protection, and then takes on a life of its own: But it was not the first function. If you are riding a horse, scooting on on a limb of a tree, walking through tall stiff grasses or brush, or for whatever working reason might have something between your legs, something to protect your genitalia is good idea for both men and women; but especially for men with more dangly genitalia.
Without a time machine to visit and ask, logic tells us the first purpose of any clothing was likely for protection (including from cold or weather), and not modesty. Obviously the "inventor" of clothing had spent a lifetime naked amongst a completely naked tribe; it is highly unlikely they began wearing some article of clothing just to hide their sexual organs. More likely, they began wearing something because it prevented injury or kept them warm.
Your gorillas can still get cold: My dog has a fur coat, but definitely starts feeling the cold and wants to get inside once the temp drops to within a few degrees of freezing.
The later functions of clothing; for decoration, status, concealment out of modesty, hiding sexual assets (like some Muslim garb) are all most likely add-on purposes due to human psychology: Once you start routinely hiding something, it becomes a mystery and attractive. It was not a lie that in Victorian times, men found even the ankles of women attractive, just because ankles, knees, legs were always concealed in public.
Our modesty is clearly a learned response, not innate. Children learn quickly, but are hilariously immodest exhibitionists in their first years, even after acquiring language. Or perhaps they are normal, and it is the adults being ridiculous.
Yes, your gorillas can have clothing. Center it on the protections they need. For example, even though we could get by without shoes, and our ancestors did for millions of years, they are so protective we do almost nothing barefoot except swimming, where they cause too much drag, and some sandy beach play activities (e.g. volleyball) where bare feet work better.
For intelligent quadrupedal gorillas, I'd start with knuckle and foot protection; they will have to work with their hands extensively (like us) and cannot afford the minor cuts, punctures, and scrapes of normal walking use.
Also, in time, by avoiding the callouses built up by knuckle-walking from birth; the gorillas may find that protection makes their hands more nimble with a more sensitive touch, which helps them work and makes their glove-like knuckle protection a necessity.
Loin protection is always a good idea; for working females breast protection is a good idea that would likely become a cultural norm. When a clothing invention (like shoes or loincloth) becomes considered such a necessity that everyone wears it, within about an average lifetime the number of people (or whatever they call themselves) that have a living memory of things being any different will be near zero; and then the rationale for wearing the item fades into the background of our mind: We wear shoes because everybody wears shoes and we were raised from infancy wearing shoes, so much so that in most situations we expect to see shoes and think something is wrong if people are barefoot: In the office, or mall, or a barefoot cop or lawyer in court. We can even feel uncomfortable about it; as if bare feet suggested disease.
Your gorillas can get cold, as they explore new territory: Much of it is very desirable for 9 months out of the year (on Earth), and bitterly cold and inhospitable in the winter. Protective clothing in these environments can help the survive the winter without leaving the place, which they could not do and protect their property: Settling down can have a lot to do with protecting property, like herds, crops you planted, ready water access, etc.
Then of course, if they are intelligent as humans, these protective purposes extend to their science (working with chemicals, high voltages, machinery, sharp or dangerous objects, etc). The functional purposes apply: Hats, helmets, glasses (to correct vision problems or protect eyes).
Then, within a lifetime or two, the persistently useful clothing becomes a cultural norm and subject to fashion. Social standing for expensive items becomes a thing, like our high fashion and expensive shoes. For us, purely functional items like a watch can become runaway prestige items: You can buy a watch to keep accurate time for around \$10, or spend five thousand times as much to wear a gold and jewels version that is actually less functional and accurate versus the cheap plastic digital version!
Another function of clothing I did not mention: Hiding body odor and uncleanliness. Obviously we have had various body odors since our beginnings, but bad smells can be signs of disease and unfitness. We have been using perfumes and clothing to disguise and conceal such odors for thousands of years, and more recently (although not world-wide) daily (or more frequent) bathing, and clothes washing (usually after a single use). I presume (without first hand knowledge) that gorillas have all the sweat and odor problems of humans.
My furred dog can get stinky. The cats don't, they keep themselves meticulously clean, likely an obsessive compulsive behavior preserved by evolution because it prevents their prey from sniffing out an ambush in the hunt; making them more successful hunters. That rationale would likely not apply to smart gorillas; and the "barrier" of fresh clothing might also help them to conceal unpleasant odors.