Yes, if you cut the nail too short, near the vein, you could hurt them and make them bleed a lot.
I found this picture that shows the 3 zones of the nail:
[link] (It's in portuguese, but it's understandable enough
)
So, the safest zone to cut it's at #3. Between zone #3 and #2 you must be really careful not to cut the vein. If you don't feel confident enough, then you should take your birds to a vet.
I've got doves too, and I tried cutting their nails once. I cut a bit higher of what I should have, and my poor dove bled a lot. I felt really bad and never tried to do it myself again.
My mom has natural talent for that, so I let her do the work.