I recently acquired a classical guitar from a relative that passed away. He had regular phosphor bronze strings on it. I switched them out for some Martin Silver-Plated Ball end Nylon strings. It sounds great but the strings will not stay in tune. They go out of tune within 10 minuets of playing. (especially that darn G string) I was wondering if the reason the strings go out of tune is because of the tension from the phosphor bronze strings. If anyone can help me with this problem it would be appreciated.
No, nylon strings take a long time to stabilize in pitch. In fact, I believe a lot longer than steel strings. So, we usually, (read, "pretty much always"), stretch our strings at the time of their installation.

Note that you can overdo that, and you wind up with dead strings from doing so.

With an acoustic, you can pretty much be certain your strings are going to lose a bit of pitch in between playing sessions, for at least a couple of weeks.

I suppose there's a fairly reliable way to know what's going on. If your new strings are getting HIGHER in pitch, then the body and neck are settling back to, more or less, where they should be. Or if you will, recovering from the excess tension of the steel strings. If the strings are becoming lower in pitch, they're stretching. Nylon is more prone to exhibit this characteristic, longer than a set of steels.

Two additional points. It is winter in the northern hemisphere. You have to be much more mindful of humidity with any type of acoustic, than with an electric. I'm reading into your screen name, and join date, that you probably have a solid body laying around. Those you can pretty much leave out on a stand all year, without much in the way of ill effect. Not so with acoustics.

This is such a thing as a slot head steel string acoustic guitar, although they're fairly rare. The primary visible difference would be, the winding pegs would be steel, whereas on a classical, they would be nylon, (or similar), material. You would also notice very narrow string slots, if the guitar was intended to be used with steel strings.

OK, good luck! I'd strongly suggest looking through this thread on the subject of humidity: https://www.ultimate-guitar.com/forum/showthread.php?t=987641

And do let us know how you make out.
In addition to what Captaincranky has said, make sure you are securing the end of the strings to the bridge in the correct manner to prevent slippage.

This should help in that regard:


Other than, consistently detuning within 10 minutes sounds a bit aggressive to simply be due to new strings, but it's not outside the realm of possibility.
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I just looked in my Oxford English Dictionary and under "Acoustic Guitar", there was your Avatar and an email address!