The first thing you should do is adjust your monitor height so it's at eye level when you're sitting with your arms relaxed. It's surprising how much that changes everything.
Like others have mentioned, posture and carpal tunnel syndrome are frequent culprits, but there's something else going on: the tingling also has a lot to do with how you tense your shoulders and neck without realizing it. I've been reading about this for years and people almost always focus on their hands, but tension built up in your cervical spine radiates down to your fingers. Try taking breaks every hour, even if it's just five minutes, and use them to drop your shoulders and gently stretch your neck.
If after adjusting your desk setup and taking regular breaks the tingling still happens or gets worse over two or three weeks, then it's worth getting it checked out. It's not cause for alarm, but it's also not something to completely ignore. Sometimes proper ergonomics sorted out from the start keeps it from becoming chronic.