There is nothing to immediately connect the "Hilkiah" the priest of Anathoth (Jer 1:1) with "Hilkiah" the High Priest, apart from their names. However, we observe the following. There are two possibilities:
1. "one of the priests at Anathoth" (Jer 1:1) applies to Hilkiah
- The High Priest was never referred to a simply a priest, so this make them unlikely to be the same person. Hilkiah the High Priest is referenced in several places 2 Kings 22 & 23; 2 Chron 34, but always as "the priest" or "High priest"
- The Hilkiah at Anathoth is mentioned as "a priest"
- The High priest had to live in Jerusalem and could not reside at a distant town like Anathoth.
Therefore, it is very unlikely that the two were the same person. The Jamieson-Fausset-Brown Bible Commentary arives at the same conclusion:
The son of Hilkiah: as this serves to distinguish him from other
priests, so his being of Anathoth ranks him among the common priests;
not that high priest mentioned 2Ki 22:8, under whose progeny Jeremiah
is not named, 1Ch 6:13; who, it is probable, would have been named,
being one of so much note, and who always lived at Jerusalem, not at
Anathoth, which was a city three miles from Jerusalem, lotted out of
the tribe of Benjamin for the priests, Jos 21:18. Of an ordinary
teacher he was made a prophet, not so the rest.
Benson arrives at this conclusion also.
2. "one of the priests at Anathoth" (Jer 1:1) applies to Jeremiah
However, it is possible that the two Hilkiahs were the same person IF we understand Jer 1:1 in a different way. If "one of the priests at Anathoth" applies to Jeremiah rather than Hilkiah, then it is quite possible that Hilkiah was the High Priest in Jerusalem and Jeremiah served as an ordinary priest in Anathoth.
The Cambridge commentary has this:
If we were to render it by Ben-Hilkiah we should no longer be in
danger of connecting the words that follow with Hilkiah rather than
with the name of the prophet himself.
Barnes, and Ellicott also have this same idea.
Finally, Gill has a much better suggestion which appears to settle the matter in favour of option #1 above:
nor is Jeremiah mentioned among the posterity of Hilkiah the high
priest in 1 Chronicles 6:13, besides, Hilkiah, a priest of Anathoth,
must be of the family of Ithamar; the last of which family that was
high priest was Abiathar, who had fields in Anathoth, 1 Kings 2:26,
and so could be no other than a common priest; for Hilkiah the high
priest was of the family of Phinehas; for, from the times of that
Abiathar to the Babylonish captivity, there was no high priest but of
that family. The Jews say that Jeremiah descended by his mother's side
from Rahab the harlot (o).