Can the President of the United States hire someone [unqualified] to become an FBI agent?
Technically, the President of the United States does not hire; but, rather appoints.
This Wikipedia article lists the positions that are filled by appointment with Senate confirmation. A 2012 Congressional Research Service study estimated that approximately 1200-1400 positions require Senate confirmation. There are 163 positions, allowing the president alone to appoint persons to these positions. None of these positions include FBI agent.
Can the director do it?
No Director would, because doing so would tarnish the agency.
Basic FBI Special Agent Eligibility Requirements
The work that FBI Special Agents perform is exacting and demanding and it requires a high caliber of individual to successfully fill the role. Because of the sensitive nature of this work, the FBI has very strict entry requirements in place. To qualify for a position as an FBI Special Agent:
Candidates must be at least 23 years old, but younger than 37 at the time of appointment.
Candidates must be citizens of the United States.
Candidates must hold a four-year degree from a college or university accredited in the United States.
Candidates must possess a valid U.S. driver’s license.
Candidates must have completed at least two years of professional work experience, or one year for those that hold a master’s or higher degree.
Candidates must comply with the FBI Drug Policy and meet the physical fitness standards described below.
Candidates must be able to be cleared for Top Secret SCI (Secure Compartmentalized Information).
Failure to maintain any of those standards, or the strict ethical and moral requirements of the job, will result in a very short career as an FBI agent.
[Emphasis added; and, the above, would apply to the Director, as well.]
Assuming the Director of the FBI were to attempt to install an individual without the necessary qualifications, that act would likely result in a report of abuse of office. (Think "whistleblower" and the fact that many of the agents, with which that individual would be in contact, are trained investigators who could spot the "scam".)
A complaint would be made to the Department of Justice, Office of Professional Responsibility.
The Office of Professional Responsibility (OPR) is responsible for ensuring that Department attorneys perform their duties in accordance with the highest professional standards, as would be expected of the nation’s principal law enforcement agency.
In addition, through investigations of FBI whistleblower retaliation complaints, OPR seeks to ensure that current, former, and prospective FBI employees are protected from reprisal when they report what they reasonably believe to be misconduct.
A report of the misconduct would be made to the Attorney General. At that point, the AG would ask the President to fire the Director.
The above process is established.
The first Director, to be fired, was William S. Sessions:
Just before Bill Clinton was inaugurated as the 42nd President of the United States on January 20, 1993, allegations of ethical improprieties were made against Sessions. A report by outgoing Attorney General William P. Barr presented to the Justice Department that month by the Office of Professional Responsibility included criticisms that he had used an FBI plane to travel to visit his daughter on several occasions, and had a security system installed in his home at government expense. Janet Reno, the 78th Attorney General of the United States, announced that Sessions had exhibited "serious deficiencies in judgment."
Although Sessions denied that he had acted improperly, he was pressured to resign in early July, with some suggesting that President Clinton was giving Sessions the chance to step down in a dignified manner. Sessions refused, saying that he had done nothing wrong, and insisted on staying in office until his successor was confirmed. As a result, President Clinton dismissed Sessions on July 19, 1993. Sessions was five and a half years into a ten-year term as FBI director; however, the holder of this post serves at the pleasure of the President.
Though not necessary to show the process, the Wikipedia article contains a suggestion that the firing may have been political.
Despite being a Republican who was appointed by Reagan, Sessions disappointed the administration of President George H. W. Bush for not being partisan, and he was personally disliked by Attorney General Dick Thornburgh. Sessions had an uneasy relationship with Thornburgh's successor William P. Barr. Reflecting the tensions between the Justice Department and the independent Bureau, Sessions announced that the FBI would be looking into whether Justice Department officials illegally misled a federal judge in a politically sensitive bank fraud case involving loans to Iraq before the Persian Gulf War, and 48 hours later Sessions was the subject of an ethics investigation on whether he had abused his office perks.