Yes, you can transfer saves and this link will give you the details:
The game saves progress when the in-game day ends, i.e., when the farmer goes to bed, collapses from exhaustion, or collapses at 2am. All progress during a day is lost if the player quits the game without ending the day. In multiplayer, the game saves after all players click through the prompts...
stardewvalleywiki.com
I do not believe you need to jailbreak anything to do this. You just need to get a particular switch app called JKSV.
Still, with Android still being on 1.4 and Switch on 1.5... an Android device can not play a 1.5 save from your Switch. A switch can take a 1.4 save from your Android and play... but as soon as you save, it'll be a 1.5 save and if transferred back to the Android device can not be played on it. This means that currently you can really only transfer one way functionally... but as soon as Android gets the update
(which we've been waiting on for a while, I will admit, there being technical and legal delays along with the creator making a big change and self-publishing, breaking off ties to the folks that were publishing the game and doing the port to mobile) you will be able to transfer back and forth.