ABSITIVELY !!! If someone brought a quo warranto action against a President-Elect in the District of Columbia [and SCOTUS accepted it], there is NOTHING to stop them from doing so. That is, unless you want to undo 208 years of Supreme Court jurisprudence - as specified in Marbury v. Madison, where the Court asserted the right of judicial review.
From Marbury v. Madison:
"The Constitution vests the whole judicial power of the United States in one Supreme Court, and such inferior courts as Congress shall, from time to time, ordain and establish. This power is expressly extended to all cases arising under the laws of the United States; and consequently, in some form, may be exercised over the present case, because the right claimed is given by a law of the United States ..."
It cannot be presumed that any clause in the Constitution is intended to be without effect, and therefore such construction is inadmissible unless the words require it ..."
"It is emphatically the province and duty of the Judicial Department to say what the law is ..."
"This is of the very essence of judicial duty ..."
"The judicial power of the United States is extended to all cases arising under the Constitution."
"Could it be the intention of those who gave this power to say that, in using it, the Constitution should not be looked into? That a case arising under the Constitution should be decided without examining the instrument under which it arises?"
"This is too extravagant to be maintained."
"In some cases then, the Constitution must be looked into by the judges ..."
The theory of judicial review in Marbury and its progeny is that it is the quintessential function of a court of law to interpret legal documents and define legal terminology, and since the Constitution is a legal document, it is a natural function of the federal courts to interpret that document.
For example, the term "natural born citizen" is a legal term used in the Constitution, and it would appropriately be the function of the federal courts to interpret that term with the Supreme Court having the final say.
To take another example, the Constitution says Congress can impeach and convict the president for "high crimes and misdemeanors." The federal courts could appropriately define this term; for example they could lawfully rule that a traffic violation was NOT a high crime or misdemeanor, and therefore the president could be made to answer for such a breach of the law in the regular courts. This is to some extent an academic point, because if the Senate were to decide that some particular crime did not rise to a high crime, even though the Supreme Court and the House of Representatives thought so, it could decide simply not to convict.
In this latter example, while the Supreme Court would have the legal power to decide what is a "low crime or misdemeanor" as opposed to a "high crime or misdemeanor" it does not have the power to try and convict a president of a high crime or misdemeanor, because that power has been committed to another branch of government under the Constitution.
Similarly, while the Supreme Court would have the final say on what the term "natural born citizen" means as a matter of law, they would not have the power to determine as a fact-finding matter that a particular president-elect did not qualify for office on that basis, as the Constitution contemplates the venue for this purpose being the Joint Session of Congress presided over by the President of the Senate.