In reading this chapter (1 Kings 10) today, I was struck by the phrase "concerning the name of the LORD" appended to "the fame of Solomon," and it impressed me that this passage deals with the queen of Sheba's conversion, not just a political enquiry.
I believe that the queen of Sheba came to Solomon as a seeker (cf. Mt. 12:42) regarding the LORD, not just to see how great Solomon himself was. Note that (1) the context of the "hard questions" that she proved Solomon with was "the name of the LORD" (v. 1); and (2) her conference with Solomon ends with her blessing the LORD (v. 9). She arrives with "spirit" (v. 5) to prove Solomon regarding the LORD, but loses that spirit after communing with Solomon (v. 2) and seeing the prosperity and glory given him by the God he worships (vv. 4-5; note the reference to "the LORD" in v. 5 along with those in vv. 1, 9). Note also Christ's reference to her in Mt. 12:42, in the context of vv. 38-42. That she was converted is implied, I believe, by comparison with v. 41, which mentions the Ninevites repentance under the preaching of Jonah. So the context here is repenting (or not repenting, in the case of Israel in Christ's day), and the queen of Sheba, which was located to the distant south of Israel (v. 42, "the south...the uttermost parts of the earth"), is included. It seems that she repented of her doubts regarding the LORD through her interview with Solomon and seeing the LORD's glory in his kingdom.
I guess, in a sense, we could call Solomon a "soul winner" (cf. Prov. 11:30, one of his own utterances) in this scene from his reign.