Developing a crush or infatuation

"Get a crush"

"Get infatuated"

"Crush on" (a newer idiom used like "I'm crushing on you!")

"Fall for"

Alternatives to "get" include, for example, "become infatuated", but I have to say if OP thinks "develop a crush" sounds too "technical", the options are probably somewhat limited.

Personally, I think infatuation sounds a bit extreme for how you might relate to fictional characters. I'd probably ask something like...

"Do you sometime experience feelings of romantic attachment to a character in a book?"

I checked on Google books, and these seemed to be by far the most common ways of expressing it with the word crush:

• Have a crush on / has a crush on / had a crush on
• Get a crush on / got a crush on
• Develop / developed a crush on

All three of those were somewhat easy to find among the many pages of hits, but I didn't manage to find any other way the words crush on were used. (I checked through at least 10 pages of the results, but all that talk about crushes started to bring up too many memories of junior high school, so I abandoned the search.)

Another term you can use for this is puppy love.