In my world there is a southern continent that features some of the fauna from Madagascar and this includes the lemur. I’m wondering if the lemurs that live on this warm southern continent could evolve or adapt to the temperate northern continent if they traveled there, potentially filling the niche of squirrels and/or something like the Japanese macaque. Could the lemurs also adapt to a subarctic or boreal climate as well?
How long do you have?
The word "adaptation" has a specific meaning in biology. Here's the definition from the Oxford English Dictionary:
A process of change or modification by which an organism or species becomes better suited to its environment or ecological niche, or a part of an organism to its biological function, either through phenotypic change in an individual or (esp.) through an evolutionary process effecting change through successive generations.
That's a fancy way of saying that, given enough generations, species can get used to a lot of things. How long do your lemurs have to adapt to their new conditions? Wikipedia has an incredibly detailed article on the evolution of lemurs that you can use for reference into the capabilities of lemurs to adapt to new conditions over long periods of time.
If you have a shorter time, your lemurs will have trouble living in a colder place. I found a 112 page guide to caring for lemurs that describes their temperature needs:
In northwestern Madagascar, there is little annual fluctuation in mean monthly temperature. The average monthly low temperature for June through August is 25 °C (77 °F), with the lowest mean temperature of 17.8 °C (64 °F) recorded during July and August. ... In very few cases, Eulemur have been exposed to temperatures as low as 4.4 °C (40 °F), but only for short periods of time. Temperatures lower than this can lead to frostbite and loss of tips of tails and digits.