According to Rashi in Melachim(Kings) 1-10:13, the words "all her desires"
refers to the following..."He had relations with her, and from her was born
Nebucadnezzar. And he destroyed the temple which stood for 410 years in the
portion of the 12 tribes." (This quote is from Rabbi Yitzchak Luria[the
'Ari'], the famed 16th century Kabbalist in Sefat.)
In the Encyclopedia of Biblical Personalities by Yishai Chasidah,
(published in English by Shaar Press, distributed by Mesorah) he cites the
same verse under the entry of Malkath Sheba(Queen of Sheba) on p. 328 and
writes, "Solomon consorted with her and she bore a daughter, from whom
issued Nebuchadnezzar, the destroyer of the Temple (Shalsheles Hakabbalah
44, citing Midrash).
I found a similar entry in the Sefer Seder Hadoros by R. Yechiel Halprin.
Toward the end of entry year 2935, he refers to the Midrash, "That this
queen became pregnant from Solomon and she bore a daughter. From her was
born Nebuchadnezzar who destroyed the Temple. And there is still a custom
in that kingdom to call their kings -David or Solomon, for they say that
they (the kings) originate from that lineage." I have not yet found a
specific statement regarding their actual marriage, but I find it highly
unlikely that he would have a child with the queen, unless she had first
converted to Judaism properly, and become his wife. I hope these sources
help to answer your question.