Short answer. Neither the Supreme Court by default, nor a Regional Trial Court — a petition involving an act or omission of a quasi-judicial agency must, unless another law or rule says otherwise, be filed with and is cognizable only by the Court of Appeals.
What the law says
If the petition involves an act or an omission of a quasi-judicial agency, unless otherwise provided by law or these rules, the petition shall be filed with and be cognizable only by the Court of Appeals.
Rule 65, Section 4 — When and where to file the petition. Read the full provision →
Different bodies, different filing courts
Section 4 does not send every certiorari petition to the same court. It sorts petitions by whose act or omission is being challenged. Where the challenge is to a municipal trial court, or to a corporation, board, officer, or person, the petition may be filed with the Regional Trial Court exercising jurisdiction over the area, or with the Court of Appeals or the Sandiganbayan. A quasi-judicial agency gets its own, more specific rule.
The rule for quasi-judicial agencies specifically
For a petition involving an act or omission of a quasi-judicial agency, Section 4 is unusually direct: the petition shall be filed with and be cognizable only by the Court of Appeals, unless another law or the Rules of Court themselves provide otherwise. The word "only" matters — this is not one option among several for this category of respondent, the way it is for a municipal trial court or a private corporation.
Why this rule is worded as an exception, not an absolute
The phrase unless otherwise provided by law or these rules keeps this from being a blanket rule with no exceptions. Some specific laws creating a particular agency, or specific rules elsewhere in the Rules of Court, can direct certiorari against that agency's acts to a different court. So while the Court of Appeals is the default and, absent an exception, the exclusive venue for a quasi-judicial agency, the correct venue for any particular agency still depends on whether some other law or rule displaces this default.
The sixty-day deadline applies regardless of venue
Whichever court turns out to be the correct venue, Section 4 also fixes the timing: the petition must be filed not later than sixty days from notice of the judgment, order, or resolution being challenged. If a motion for reconsideration or new trial was timely filed, that sixty-day period instead runs from notice of the denial of that motion. Getting the venue right does not extend this deadline, and missing it can defeat the petition regardless of which court would otherwise have been correct.