Landforms like this can be aeolian (i.e. created by wind), alluvial (i.e. created by water) or glacial (i.e. created by movements of large masses of ice).
For example, there is a line of hills that run across the midwest (south of Columbus and north of Cincinnati in Ohio) that is a terminal moraine where a glacial dumped debris it scraped up when it melted at the front of the advancing glacier. Viewed end on, a terminal moraine can look like a single peak.
Craters and masses of atypical substance in a planetary crust can also be created by extraterrestrial impacts (e.g. meteors and comets). For example, suppose at a very large iron meteor hits a planet long in the past burying itself deep in the earth and then erosion by wind and water gradually erode the area around it, voila, you have a mountain. It may not be a really awesome mountain by the standards of places that have volcanic activity. But, in a place where everything else is pancake flat, it could still be pretty impressive.
As yet another possibility, you could also have a very tall, bushy tree with a deep root network the starts the foundation of the mountain that gradually collects dirt that ossifies over time.
The pile wouldn't have to be botanical either. The white cliffs of Dover are an accumulated pile of seashells that could take a mountain shape if carved by rivers on either side.
Also, while it isn't clear if you merely want no volcanic activity or really mean no tectonic activity, colliding tectonic plates can create mountains that are not volcanic:
The Himalayas, the European Alps, and the New Zealand Southern Alps
are all examples of non-volcanic mountain chains caused purely by
collision of continental crust at plate boundaries.