I take the question charitably, as offered in good faith. Instead of a "generic" Gentile couple, let us assign (for the sake of offering an answer) the identities of the Philippian Jailer, with his wife and child. I'm not asking the Baptist to accept a priori my take on the events of Act.16; but merely to put the hypothetical into something better than a purely imaginary scenario.
If, indeed, what happened that night was right in line with the hypothetical, we do not need (on one hand) to wonder what relevant matters were spoken, which led to such obedience as household baptism, including an infant. The text contains "the minimum" required.
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But, perhaps we can imagine additional requests for reasoning from the Holy (OT) Scriptures--as has been mentioned they are the only Bible in existence--the way Paul might have explained himself.
Father Abraham--who according to Paul's gospel is the father of ALL the faithful, of Jews and of Gentiles--he was the first to be directed to mark as God's ownership his whole household (the ones who were capable of wearing the mark). "I will be God to you, and to your seed after you" (Gen.17:17).
We know the apostolic preaching--of which we have several epitomes in Acts--included references to the ancient promise (Act.13:32), phrasing that encompasses fully all the particulars of promise as they came to one patriarch and one OT saint after another across the sacred record. Peter in the first case of apostolic preaching makes explicit connection of the present conditions to the Abrahamic promise, Act.2:38-39.
The promise included the calling of the Gentiles, Gen.12:3; 17:4; Paul's preaching included references to such, as Act.13:47; cf. Is.49:6. Remember now, this is not replacement theology; but rather inclusion theology. The Gentiles are to be offered seats in the kingdom feast beside Abraham, Isaac, and Jacob; see Mt.8:11-12. These are branches grafted on the Vine, Rom.11:17. The promise is given to believers, without respect (intrinsically) to heritage.
"The promise is to you, and to your children...," declares Peter in the NT context, echoing the original OT wording (and to the Gentiles "afar off"). And if Abraham's infant sons should also receive that sign, of the faith they should also exercise themselves (if they would be true sons); then so much more the sons--and daughters--capable of wearing the mark should wear it in the NT age. Since God had put such in his visible kingdom from the beginning, and nowhere ordained their exclusion or removal, then the same remain in his visible kingdom.
So far is the logic of infant-inclusion. Isaiah, in reference to the days of the New Covenant writes, 59:20-21, "And the Redeemer shall come to Zion, and unto them that turn from transgression in Jacob, saith the Lord. As for me, this is my covenant with them, saith the Lord; My spirit that is upon thee, and my words which I have put in thy mouth, shall not depart out of thy mouth, nor out of the mouth of thy seed, nor out of the mouth of thy seed's seed, saith the Lord, from henceforth and for ever;" and two vv along, 60:3, "And the Gentiles shall come to thy light, and kings to the brightness of thy rising." It is hard to escape the observation that the OT envisions the future Spirit-age as including even "nursing daughters," v4.
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Does Christ, by means of his servants, supervise a visible manifestation of his kingdom in this world? Does the NT age continue an administration of the Covenant of Grace by way of this-worldly signs and exercises? I know many Baptists refuse this admission. They say no, but only the Holy Spirit administers the (new) covenant by invisible workings. So, at this juncture I'm not sure if there is an appeal to the OT Scriptures, or the NT, that will persuade the convinced Baptist that the church is the embassy of the "far country" (Lk.20:9), a visible exhibit of the kingdom of God and its administration for saints on the way.
Baptism, we teach, says essentially the same thing that circumcision once declared; the differences though real are of a second-order. The sign is initiation into the life of discipleship. Neither sign has ever been effectual apart from faith in what is signified. Again, in distinction from the primary sense of (most) Baptists, we do not think of baptism first of all as man's witness unto God and man of his personal faith; but that baptism happens to a man, it is symbolically monergistic though it elicits the public witness in response, it is God's declaration: "I will save even this one, who believes in me."
Amen, so let it be.