There are some parts of the Bible like Psalms, which are mostly songs or poems. They don't rhyme in the currently translated English obviously, but I'm curious if the original, untranslated songs or poems had rhyming in them.
Not normally, no.
Rhyme is predominantly a feature of English poetry.
Hebrew poetry, as in the psalms, is marked by other features. These include parallelism (including synonymous parallelism, where two lines express a similar or singular idea; antithetical parallelism, where two lines express a contrast; and a whole range of others); possibly accentual rhythm, though this is strongly debated; acrostics are used in some of the psalms (starting each section with the next letter in the alphabet).
Basically, characteristics of 'poetry' differ from language to language, and period to period (indeed, English poetry was not originally so dominated by rhyme, but by other patterns).
While u/talondearg is not incorrect, the more accurate answer is that we do not have an accurate enough understanding of the actual phonetics of Hebrew to be able to make definitive statements about rhyme scheme.
However, that said, we do have places where it seems that poetic tools like assonance are at play (Ps 29:9, for example, note the repetitive /รด/ vowel replicating thunder in the Hebrew text), and, elsewhere, alliteration (although no good examples come to mind immediately).
The best resources on Hebrew poetry are going to be James Kugel (The Idea of Biblical Poetry, retired, formerly Harvard prof.), Chip Dobbs-Allsopp (Princeton prof.), and Jackie Vayntrub (currently at Brandeis).