OP: Which prophecy[ies] might suggest, more specifically, that "the time had come" for the Messiah to be born?
The idea that there was a very specific time for Messiah to appear is very clear from Scripture. For example, Paul mentions it this way.
But when the fulness of the time was come, God sent forth his Son, made of a woman, made under the law, Gal 4:4
The word "fulness" means this.
Of time (see πληρόω, 2 b. α.), that portion of time by which a longer antecedent period is completed; hence, completeness, fullness, of time: τοῦ χρόνου, Galatians 4:4; τῶν καιρῶν [-source-][1]
Another example of God’s set time table is from Luke when Pentecost had fully come when the Spirit was poured out (Acts 2:1).
So, to what exactly does that "fulness of time" point? To help, let's ask within the context, how could something be full without a known starting point? Obviously, some had to know when that was.
There is only the well-known Daniel 70-weeks prophecy that nails down both the start and end points to when Messiah HAD to appear. Of course, the prophecy has various interpretations, but the point is it clearly prophesied that Messiah had to appear, be anointed, and cut off.
For the OP, the question is about His birth specifically, but to detail this, we need to consider the whole prophecy. We need to understand the anointing portion of the prophecy. Some believe this was when He was born, but the language is clear enough that it is about His anointing; to wit, His baptism at age 30 into the priesthood.
24 Seventy weeks have been determined upon thy people, and upon the holy city, for sin to be ended, and to seal up transgressions, and to blot out the iniquities, and to make atonement for iniquities, and to bring in everlasting righteousness, and to seal the vision and the prophet, and to anoint the Most Holy.
25 And thou shalt know and understand, that from the going forth of the command for the answer and for the building of Jerusalem until Christ the prince seven weeks, and sixty-two weeks; and then shall return, and the street shall be built, and the wall, and the times shall be exhausted. LXX Dan 9:24-25 bolding mine
There’s the definition that Paul worked with. The fulness of time is 7 plus 62 or 69 weeks or 483 years (using a solar year) measured from the command to Christ. But was it to Christ’s birth, death, or something else?
26 And after the sixty-two weeks, the anointed one shall be destroyed, and there is no judgment in him: and he shall destroy the city and the sanctuary with the prince that is coming: they shall be cut off with a flood, and to the end of the war which is rapidly completed he shall appoint to desolations. LXX Dan 9:26 bolding mine
To make atonement mentioned in verse 24, it requires a priest and sacrifice. Christ fulfilled both. To the point, in verse 26, He is called the anointed one. When was He anointed?
The Spirit of the Lord is upon me, because he hath anointed me to preach the gospel to the poor; he hath sent me to heal the brokenhearted, to preach deliverance to the captives, and recovering of sight to the blind, to set at liberty them that are bruised, Christ quoting Isa 61:1
This was at His baptism.
. 16 And Jesus, when he was baptized, went up straightway out of the water: and, lo, the heavens were opened unto him, and he saw the Spirit of God descending like a dove, and lighting upon him: 17 And lo a voice from heaven, saying, This is my beloved Son, in whom I am well pleased. Mt 3:16-17
From the issuing of the command until the anointed one as a priest is 69 weeks. Subtract 30 years from the 483 years (69 weeks of years), the age when men may become priests, and we get His birth period. This age requirement may or may not apply to those of the Melchizedek priesthood, but it certainly applied to John the Baptist of the Levitical tribe who was the forerunner. This, by the way, is one reason they thought John might be the Messiah.
So, from the issuing of the command in the year 458-7 to 453 years (483-30) later and Messiah must be born in 5 BC.
To answer the OP, this is how some knew the time of His birth because the fulness of time had arrived from which it could be measured from the initial command.