Not necessarily.
The authors of the Bible worded what they said very carefully. Paul included. The fact that he DID NOT mention fits very well with what he said about members of the Body of Christ being "neither Jew nor Greek."
The holy priesthood of Israel was, by definition, of the Jews. Therefore, "neither Jew nor Greek" excludes any form of priesthood.
In addition, every mention of priests and priesthoods (apart from instances where it's talking about pagans, and excluding the references to the Order of Melchizedek in Hebrews) has to do with Israel specifically, and that fact is made obvious by the context. In fact, aside from the several hundred mentions of priest, priests, priesthood, etc, in the Old Testament, there are 26 mentions in Matthew (with one questionable mention that is likely only referring to pagan priests in Herod's court near the beginning of Matthew) that specifically refer to Israel's priesthood, 22 mentions in Mark, 23 in Luke, 21 in John, 24 in Acts (the 25th being a priest of Zeus, and thus is one of the exceptions mentioned above), about 23 in Hebrews with a few of those questionable, but only because they could be referring to Israel's priesthood, priesthoods in general, or the priests of the Order of Melchizedek, 2 in 1 Peter, and 3 in Revelation, all of which refer to Israel in some way or another.
None of those examples, even including the ones that aren't specifically about Israel, mention anything that would be at all relevant to the Body of Christ, which Paul describes as a creature, an organism with many members, and not a nation of priests.