The answer is pretty simple: Quintus as a name can be both a Praenomen (somewhat similar to our given name) or a Cognomen (somewhat, but not quite, similar to a nickname, though usually hereditary and used to differentiate between different family groups or individuals), originally probably referrring to being born in the fifth month, while Primus, Secundus, Tertius and Quartus were only used Cognomina, meaning they would naturally occur less often.
Since the Romans only every used a very narrow range of Praenomina, one of which was Quintus, it would consequently occur more often, especially in republican times, when Cognomina weren't as common. Since the Cognomina like Secundus, Tertius and Quartus often referred to the order of birth, they were progressively less often used, since even back then not everybody had four or five sons. You can also see that in the inscriptions that mention the respective names, if you search the databases you find Primus most often and Secundus, Tertius, Quartus each less often than the one before.
It doesn't only seem that way: Quintus as a name did occur more often. You can find (using the Database Clauss-Slaby and searching only for the nominative) 7325 results for Quintus, 2092 for Primus, 1810 for Secundus, 711 for Tertius, 208 for Quartus.
see also: how do Roman names work, as in this question
or this one
The answer posted by u/Astrogator is true, but we can go further. The answer may astonish you. It certainly astonished me when I found it out a couple of years ago.
The true reason is this: some Roman personal names, including Quintus, come from month names. There are no month names corresponding to Primus, Secundus, Tertius, and Quartus, so there are no personal names with those numbers.
This theory was put forward by the Roman scholar Varro, and it seems he was dead right. The relevant modern study is Petersen 1962, which hasn't been superseded or refuted.
Some caveats:
Here's the full set of names that come from month names. A praenomen is the first name, gentilician name is a family name. (There are also cognomina, surnames that go after the family name, and for those anything is fair game -- including Primus, etc. as u/Astrogator points out.)
| Latin month name | Praenomen | Gentilician name |
|---|---|---|
| Martius | Marcus | Marcius |
| Aprilis | -- | -- |
| Maius | Maius | Maius |
| Iunius | Iunius (very rare) | Iunius |
| Quin(c)tilis | Quintus | Quinctius |
| Sextilis | Sextus | Sextius |
| September | Septimus (rare, archaic) | Septimius |
| October | Octavius (rare, archaic) | Octavius |
| November | Nonus (rare) | Nonius |
| December | Decimus | Decius (Roman), Decimius (Sabine) |
First thing to notice is that there are no names corresponding to January, February, or April. This suggests a very archaic origin for this set of naming customs:
January and February seem to have been added to the start of the canon of months at some point. (Notice that the number-related months don't correspond to their correct place: you'd imagine Decem-ber would be the tenth month.) This is what the Romans themselves believed, and they were probably right. They also believed it was the legendary king Numa that added January and February: that bit is almost certainly false. Still, it was early. Since there are no personal names corresponding to those months, the naming custom is probably earlier than the addition of January and February. (Whatmough 1931: 171 suggests that February was linked to Juno under the titles Februata and Lucina: if the month was transposed from the end of the calendar, or if one of these titles was formerly attached to the intercalary period before March, that could provide a plausible origin for the praenomen Lucius.)
The month name of April is probably an Etruscan borrowing, from afr, apru(n), 'fortune, (goddess of) Fortune'. It's possible there was a native Latin name prior to that borrowing: if so, we could well expect that another praenomen corresponded to that lost month name: a good candidate would be a name related to a major divinity (cf. Mars > Martius > Marcus), or a praenomen + gentilician name pair that follows the same pattern as some of the others, like Gnaeus/Naevius.
The main argument in favour of Varro's theory is that the number-related praenomina go from 5 to 10, just as with the number-related months. Petersen cites a bunch of parallels from two languages related to Latin, Faliscan and Oscan. We get Mamerkis (< Mamertti- 'March', Mamertis 'Mars'), Sepis ('seventh'), and Decis ('tenth') as Oscan personal names. But one Oscan family name, Sehsimbriis, is harder to interpret: the -m- infix implies a month name 'Seximber', which seems doubtful to me (though Whatmough 1931: 174 accepts it). The family name Florus could potentially be related to a northern Oscan name for July, Flusaris. The Roman name Pomponius probably comes from Oscan pumperias, pompe 'five' (*k^w > Oscan p, Latin qu). But we've also got the name Petronius, probably from Oscan pettiur, pitora 'four', and that doesn't follow the same pattern as the Roman names. It could be that the Oscan month did have a name relating to 'fourth' -- we know it was called something relating to June in the towns in the immediate vicinity of Rome, but further afield we don't know what people called it -- but that has to be just a speculation.
References