Short answer. With the Court of Appeals. The rule provides that where the petition involves an act or omission of a quasi-judicial agency, it shall be filed with and be cognizable only by the Court of Appeals, unless otherwise provided by law or the rules.
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 →
The word 'only' is doing real work
Elsewhere in the same section the rule offers alternatives — a petition against a municipal trial court, a corporation, a board, an officer or a person goes to the Regional Trial Court of the proper territorial area, and may also be filed with the Court of Appeals or the Sandiganbayan. For a quasi-judicial agency the drafting changes. The petition is to be filed with and cognizable only by the Court of Appeals. There is no menu here, which is exactly why the classification of the respondent matters so much.
Everything turns on whether the body is quasi-judicial
The rule does not list which agencies qualify, so the practical question becomes whether the particular body was acting in a quasi-judicial capacity at all. Government offices do many things — they issue permits, adopt regulations, run internal administration — and not every act of every agency is of the kind this clause addresses. Getting this wrong sends the petition to the wrong court, and the periods in this rule are short enough that a misdirected filing is a real risk rather than a recoverable detour.
The rule leaves room for other laws
Note the qualifier: unless otherwise provided by law or these rules. The exclusive assignment to the Court of Appeals is the default, not an absolute. Some agencies are governed by their own statutory review arrangements, and where a law provides otherwise, that provision is what the rule defers to. So the enabling statute of the agency you are dealing with is part of the answer, alongside this section.
Mind the period while you settle the forum
The same section sets sixty days from notice of the judgment, order or resolution, and provides that where a motion for reconsideration or new trial is timely filed the sixty days run from notice of the denial of that motion. Those periods keep running while the forum question is being worked out. The sensible order is to fix both together — where it goes and by when — with a lawyer, as soon as the ruling comes in, rather than resolving one and discovering the other has closed.