If you have been placed on formal felony probation for a drug crime - NOT DIVERSION or PROP 36 - you absolutely have been convicted. And it will show on your record as a felony conviction. The truthful answer would be an unqualified "yes."
As you know, one of your conditions of probation is not to commit any new crimes.
But not every lie is a crime. If that were the case, most of our elected leaders would be officially treated as criminals. Perjury is one classification of crimes involving dishonesty, but it requires that the lie be under oath. If your rental agreement required you to take an oath and/or sign under penalty of perjury, then you would be guilty of a crime.
All fraud crimes are crimes of dishonesty, but typically require a specific intent to deprive another of money, use of property, or services through deception or impersonation of another. For instance, it is illegal to commit insurance fraud ($$$), mortgage fraud ($$$), fraud to obtain telephone services (services), pretending to be someone you are not (cop, firefighter, state employee, licensed professional, etc.) (deception), identity theft, etc. I do not know of any criminal fraud statutes that covers a misrepresentation on a rental agreement.
False statements can constitute a crime depending on the intent and circumstances - false financial statements, false statements to DMV, false statements to the cops, etc. While I have not done an exhaustive search of every statute and ordinance, I know of no law that makes lying about your criminal background on a rental agreement a crime. Of course, your lie will invalidate the lease and will permit immediate termination of your lease once discovered.
Obviously, nothing said here is meant to encourage you to lie. And there certainly may be civil liability that flows from a deliberate misrepresentation. Also, depending on the language of your other conditions of probation, a lie like this could conceivably violate one of those conditions.
Probation is not a conviction; it is a grant of leniency. Under the terms of any probation your imposition of your sentence is suspended or execution of your sentence is suspended. There should be no felony conviction on your record, but if they check criminal records there will be a record of your arrest and the disposition in your case.
This leaves you between a rock and a hard place. If you answer “yes,” that is not exactly accurate and you won't get the apartment. If you answer “no” and they run a criminal records check, the arrest will show up and many landlords would reject the application for that reason.
IMO the best course is to tell the truth with an explanation. Not every landlord is so hard-hearted that you'll be turned down. If they were, everyone who is now losing their homes to foreclosure would be on the street.