One AHD online (if my copy/paste works):

brook1 (brʊk) pronunciation
n. Chiefly Northeastern U.S.

See creek (sense 1). See Regional Note at run.

[Middle English, from Old English brōc.]

brook2 (brʊk) pronunciation
tr.v., brooked, brook·ing, brooks.

To put up with; tolerate: We will brook no further argument.

[Middle English brouken, from Old English brūcan, to use, enjoy.]

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Apparently this indicates that “brook” and “brooked” have different etymologies.  Is that correct?

Yes they do, droogie.

The one that means “creek”: Middle English ‘brook’, ‘broke’, from Old English ‘br(long o)c’; akin to Old High German ‘bruoh’ marshy ground and probably to Old English ‘brecan’ to break

The one that means “to put up with”: Middle English ‘brouken’ to use, enjoy, digest, from Old English ‘br(long u)can’; akin to Old High German ‘br(long u)hhan’ to use, Gothic ‘br(long u)kjan’ to use, partake of, Latin ‘frui’ to enjoy

(from Merriam Websters Unabriged) http://unabridged.merriam-webster.com/