If it's for a book with a print publisher (large or small press) who paid you an advance, you can ask your editor/agent if another writer(s) in the same house can do a quote for you. Usually editors are more than pleased to promote within the house.
If it's a self-pub, vanity, e-book--anything without an advance involved--forget it. Unless you are personal friends with a published pro, they are going to be too busy for you.
You can ask a quote from other writers in the same venue if you think they might have the time, but it won't be worth much except to fans of that venue.
Keep in mind that the higher up the ladder the writer is, the less time she has to read to offer a quote.
I asked half a dozen writers if they might have time to read my latest and offer a quote before I sent it off to my editor. Four were happy to help out, one I've not heard back from, the other said he was just too busy and wasn't giving quotes out to anyone this year. Considering his schedule, I don't blame him!
Approaching another writer you admire who is not in your house is just a matter of being polite.
"My new novel __ has been sold to __ Press, and I was hoping you might be able to do a cover quote for it. Please let me know if it's possible, and I'll be glad to send you an ARC."
Mentioning you're a big fan may not help, but it never hurts. Make sure you ARE a big fan.
My ARC was in what I call "magazine format" printed at the local copy shop. The whole book was single-spaced, double columns, printed both sides of the page, 11 point Times Roman font with index stock covers and coil bound. It reduced 450 pages to 76 pp. and was a lot easier to read and mail. My version was autographed with a big thank you to the writer inside.
I could have sent e-copies, but those files are too easy to not open, then forget about as new emails come in. I've one sitting in my IN box for a book a publisher sent me over a year ago. I just forgot about the danged thing. It's probably in print by now, I just don't know.
These went to writers I've worked with and edited. If I'd never met them, I'd have had my agent or editor ask for me.
If they say no, don't take it personally. It just means they're too busy.
