In effort to transition my mountain bike to a city commuter- can I replace the mountain tires with thinner city tires without changing the bike wheels?
Short answer: Yes, you can.
The rims on most mountain bikes will support a tire as thin as 26 x 1 inch. 26 x 1.25 is virtually guaranteed on any mountain bike at all. (As long as it is a 26" wheel bike.")
In my opinion it's a fallacy to believe that a city bike should have thin tires and in fact if you look at bikes that are specifically marketed as city bikes (and don't aim at the hipster single speed crowd) they often come with quite wide tires.
Wide tires offer much more dampening and the difference in rolling resistance is much smaller than once believed. Since city roads and bike lanes often aren't very smooth this means a considerable plus in comfort at virtually no cost.
For me slightly modified MTBs make perfect city/allround bikes and as far as tires go I personally love Continental Town & Country.