Is it possible to change the bike’s friction shifter into a trigger shifter?
Yes, this is possible. You just have to select trigger shifters that are compatible with your derailleurs.

As far as as know, Shimano only offers twist shifters in their low-level Tourney range, which are all rear 7 or 6 sprockets, and either triple or single chainring cranks. You just need Tourney shifters that match the number of sprockets in the rear and chainrings in the front. There are several models of Tourney shifters but they are all interchangeable.

The same principle should hold if you SRAM gripshift twist shifters and derailleurs.