I have to translate this into Serbian, and if I'm not wrong, it goes like this:
dilber - dragan, voljeni, miljenik, lepotan (darling, dear), but here I think it refers to a beautiful woman, a belle.
hurme - urme? (fruit dates)
momče hurmama - momče urmi ? (a boy, chap of dates ?) I think this has to do with Islam or Ramazan holiday...
sabah - jutro, svitanje (morning, morn, dawn)
sane -> san; sni, snovi ??? (dreams) or sleepy, drawsy
aj sane zaspala dilbere Sajma u bašti
The belle Sajma fell asleep in the garden
aj budilo je mlado momce hurmama
A young (noble) boy was waking her up (from a deep sleep)
"Aj ustaj Sajmo, aj ustaj zlato, sabah zora svanula!"
"Come on Sajma, get up, get up darling, the morning has broken!"