Can a fugitive make a pleading in court to quash the warrant for their
Sort-of yes, but in practical terms no.
- A fugitive cannot seek the assistance of a court of which they are in contempt, but that is unlikely to apply. In any case, it is discretionary.
- There is probably no rule against such a pleading.
- A lawyer would be ethically prohibited from filing such a pleading, because they cannot represent a fugitive while he or she remains at large.
In the scenario of interest, the fugitive has not been indicted. That means that the arrest warrant must have been issued in a stand-alone proceeding, as distinct from a bench warrant (which is only issued if the accused fails to appear at a hearing and is therefore found to be in contempt).
I am also assuming that the fugitive is hiding, not merely out of reach. That is, they are probably still within the jurisdiction but the authorities can't find them. The situation is different if the authorities know where the fugitive is but they are out of jurisdiction e.g. Julian Assange.
Where a person, having been convicted of an offence and filed an appeal, subsequently goes on the run, the court may (in its discretion) dismiss their appeal on the basis that the appellant is no longer under the control of the court: Smith v United States (1876) 94 US 97. This is similar to the rule that a court may refuse to hear applications by a party who is in contempt of that court and/or has indicated that they will disobey the court's order if it is not favourable to them. The underlying principle relates to a court's inherent power to protect its own dignity and foster the obedience of its orders.
However, the fugitive dismissal rule only applies where the person goes on the run during an appeal, not if they go on the run before filing for an appeal: Ortega-Rodriguez v United States (1993) 507 US 234. By analogy, it would probably not prevent a person from filing a writ while a warrant for their arrest is outstanding.
The outcome may be different for a bench warrant. In such a scenario, the fugitive is in contempt of the court to which they may be applying to challenge the warrant (although they may be applying to a different court, e.g. to a federal court to challenge a warrant issued by a state court). In such a case, the court may exercise its discretion to refuse to hear applications on behalf of the fugitive until they submit to the court's jurisdiction by surrendering themselves.
Talking more generally (covering the scenario where the fugitive dismissal rule does not apply), the issue of whether the party can attend court without being arrested usually relates to their ability to give evidence as a witness (rather than their ability to actually file a claim or a defence).
For example, Roman Polanski, who has been a fugitive from US authorities for some time, sued a magazine for defamation in 2005 in the UK. He had no problem getting his claim filed through his lawyers but ran into issues when the success of his claim required him to give evidence himself. Ultimately he was allowed to give evidence by video link because by entering the UK he would risk being arrested and extradited to the US. The House of Lords held that to deny him the opportunity to give evidence by video link would deny him any access to justice: Polanski v Conde Nast  UKHL 10. (The tension between the self-inflicted nature of fugitive status and the principle of access to justice is discussed at, among other places, paragraphs 30 and 31.) You are talking about quashing an arrest warrant, not defamation, but they're both civil proceedings and the principles are the same.
However, another issue in this scenario is the fact that a lawyer cannot, ethically, represent the client while the client remains at large. There are conflicting opinions on whether a lawyer, contacted by a fugitive, is bound to disclose the fugitive's location to the authorities or whether this information is privileged. Such a scenario raises a conflict between the lawyer's duty of confidence to the client and their duty to the legal system. However, everyone agrees that a lawyer must, even if they decide to maintain the confidence of this information, advise their client to surrender themselves and withdraw from the case if they refuse to follow that advice. See e.g. https://www.gabar.org/barrules/handbookdetail.cfm?what=rule&id=465.
Therefore, while there is generally no rule formally prohibiting a fugitive from challenging a warrant for their arrest while they are at large, in practice they could not do so because they cannot attend court in person nor can they get a lawyer to do so on their behalf. Therefore, a fugitive must surrender themselves and then try to sort out any errors in what they have been accused of. As it went in The Fugitive, 'I didn't kill my wife' ... 'I don't care.'