I heard that Ibn Ezra preserved some of Solomon Ibn Gabirol's exegesis in his great commentary on the Pentateuch, but did Ibn Gabirol write a commentary on King David's Psalms?
There's a disagreement whether or not Ibn Gabirol authored a commentary on the Bible:
On one hand:
The Stanford Encyclopedia of Philosophy wrote (emphasis mine):
"We also have evidence in other medieval Jewish authors of what might be part of an allegorical commentary on the Bible by Ibn Gabirol (as found in Abraham ibn Ezra's commentary on the Bible; see Friedlaender 1877, p. 40; Kaufmann 1899, p. 63ff.; for English see Sirat 1985, p. 79)..."
"In addtion, he [Ibn Gabirol] composed Bible commentaries, fragments of which are cited by R. Abraham ibn Ezra in his own commentaries.
The passages from Ibn Gabirol's commentaries cited by Ibn Ezra do not contain any formulation of a theory of biblical exegesis, but it is clear from them that Ibn Gabirol's central tendency was to harmonize between philosophy and the biblical text by interpreting these texts as neo-Platonic philosophical allegory..."
But on the other hand:
The Jewish Encyclopedia wrote (emphasis mine):
"Some specimens of Gabirol's skill as an exegete are preserved in the commentaries of Abraham ibn Ezra (comp Bacher, "Bibelexegese," pp. 45-55); idem, "Ibn Ezra als Grammatiker," p. 183; and Bárány," Salamon ibn Gabirol mint Exegeta," 1885, pp. 10-17). It is not known whether Ibn Ezra cited these exegetical passages from a Biblical commentary of Gabirol, to which work there is no extant reference, or from a special work devoted to Biblical exegesis."
"While it would be ideal to look to the Bible commentary by Ibn Gabirol himself to see what his stance was...we are not able to do so. Selections of his Bible exegesis have survived to the modern day, but only in citations that appear in the works of other medieval scholars...Indeed, it is not clear if Ibn Gabirol wrote a commentary to the Bible in its entirety or only to specific sections..."
"Gabirol’s literary activity may be classified under the following headings: Biblical Exegesis, Grammar, Philosophy and Ethics, and Poetry. That he actually wrote a commentary on the Bible is doubtful, but there are indications that he did not neglect the Bible entirely. Abraham ibn Ezra cites him on three occasions in his commentary on the Pentateuch, once in his commentary on Isaiah, twice in his commentary on Psalms, and once in his commentary on Daniel. All these instances, however, are examples of the allegorical method of interpretation, and it is possible that they were taken from some philosophical work of Gabirol or from some special work on the subject of biblical allegory. In fact we have two citations from Gabirol in David Kimḥi’s commentary on Psalms (37-8, 23) which are taken from his ethical work. But the biblical illustrations in which this very work on ethics abounds lend countenance to the suggestion that Gabirol also engaged in biblical exegesis."
In short, though there may have been a commentary on the Bible or at least parts of it, there doesn't appear to currently be any evidence that there was one specifically on Psalms.