Nope, does not sound a very legit claim - I'm trying to think where your lecturer would have gotten that idea, but any ancient source definitely doesn't say anything like that. I've written previously about the relationship between Alexander and Aristotle, and basically we don't know anything at all what Aristotle taught and said to Alexander. Aristotle taught Alexander only for a year or two when the prince was 13-14-years-old, although it's plausible that they corresponded until Alexander's death. Aristotle's and Alexander's teacher-pupil relationship was glorified in later antiquity and medieval times and people wrote things such as 'fake correspondence' between them (e.g. Secret of Secrets) - I am not familiar with these works but it is possible that your lecturer has gotten the story from some fictional work.

I also don't think Aristotle would have known that boiling water can make bad water drinkable. Aristotle ascribed to the Hippocratic idea of disease, i.e. that there were four 'humors' in the body, melancholy, choleric, sanguine, and phlegmatic. They corresponded with the four primary fundamental 'qualities' in life: hot, cold, wet, and dry; and the four 'elements': earth, air, fire, and water. They were respectively represented in the body by black bile, yellow bile, blood, and phlegm. Sickness was caused by the imbalance of these four humors in the body. Hippocrates writes in great length in his On Airs, Waters, Places how different kinds of waters (e.g. salty, hard, cold) have different effects on your health; for example, water from marshlands contains too much black bile and therefore disturbs the balance of your body, which can make you ill. Hippocrates believed that the geography and topography of the water source is what made the water good or bad, and he does mention that certain waters might need boiling because of their 'structure':

Such , [rainwaters] to all appearance, are the best of waters, but they require to be boiled and strained; for otherwise they have a bad smell, and occasion hoarseness and thickness of the voice to those who drink them.

But, he definitely does not say anything to the effect that all unknown water should be boiled or that boiling water can prevent disease. Looked up also what Aristotle wrote in On Meteorology 4 about boiling, but it's all about theoretical questions such as what happens when things boil, what can boil, why different substances react differently to boiling etc., nothing about preventing diseases.

Followup question: Did any other ancient people make the connection between boiling and water potability? It doesn't seem like you would need to actually understand germs to make the connection.