You sure can!
9x2 = 127 square inches
9x1.5 = 95 square inches
6x2 = 56 square inches—> 44% of 9x2, 58% of 9x1.5
So, you’d want to make 44% of a 9x2 recipe for a 6x2 recipe and 58% of a 9x1.5 recipe, but I’d bet you’d be fine rounding off to 50% or 60%, respectively.
Also, since the diameter of the pan is less, the cake doesn’t need as much structure to hold itself up across the span of the pan as it does for a 9x2, so if you want a flat layer, you’ll want to weaken the structure by adding a bit more Baking Powder. This is a guess: add 1/16 tsp for a 6x2. However, if you don’t mind a dome, you can likely leave the BP alone. One of our members, Jenn, routinely makes cakes in the 6” size, and I don’t think she uses any leavening modifications.
For cooking time, most take 30-40 minutes. I’d probably look through the glass at 20 minutes, just to see what’s up. If you bake toward the low end of the time scale, it could be done. If yours are done toward the long end, it might not be. But, chances are, you can use whatever time works for your first cake as a fairly accurate go-by for other similar cakes.
I can usually look through the glass and tell when it’s worth “the touch test.” All hits of moisture gone from the top of the cake and a sort of look of stability. Usually, I’m a little early and it doesn’t spring back, but it helps me judge when I want to check it next, usually 3 minutes.