No, and yes. Some chilis are climacteric and some are non-climacteric, for example Habaneros are non-climacteric (source: https://academic.oup.com/plphys/article/159/4/1713/6109558).
Unfortunately, with chilis it's complicated. Some sources say that some chilis are climacteric early in their life-cycle, but become non-climacteric after being picked. It may also depend on when you pick them. Finally, as other research papers have mentioned, the specific climacteric effects in chili peppers vary. Sometimes it causes changes to flesh thickness and softness, but no changes to color, etc.
On top of that, chili-specific information online is very inconsistent; from some claiming that putting chilis in a paper bag will continue ripening to people claiming it never works. Some say chili A is climacteric, others say it isn't.
In terms of spiciness, most sources claim that capsaicin (spicyness) will not improve after being picked, in most chilis.