There’s actually a Wikipedia page dedicated to the phrase!

Relevant section:

While the earliest documented use of the expression remains somewhat nebulous, it is generally regarded as having been coined by Samuel Taylor Coleridge. Published in 1825, Coleridge’s first verse in the poem “Work Without Hope” refers to both bees and birds in reference to the coming fecundity of spring:

All Nature seems at work. Slugs leave their lair—
The bees are stirring—birds are on the wing—
And Winter, slumbering in the open air,
Wears on his smiling face a dream of Spring!

One scholar notes an earlier reference to “birds and bees” on columns in St. Peter’s Basilica from a 1644 entry in the diary of English writer John Evelyn. By the late 19th century, the phrase was common enough to appear in such works as essays by John Burroughs and publications explaining reproduction to children.

The sources for the entry go into further detail: https://web.archive.org/web/20210510050626/https://www.latimes.com/archives/la-xpm-2000-sep-04-cl-15141-story.html and https://www.livescience.com/39316-birds-and-the-bees.html