My partner wants to separate, and wants me to pay spousal support. We live together and aren't separated yet. Is this legal?

No.  Your partner is misinformed.  If you have not yet separated – and in fact are still in a customary living-together arrangement (meaning  one that would not be characterized as “separated but living under the same roof), then your partner has legal basis for claiming support from you at this time.

Support entitlements (if any) between former partners arise only upon legal separation, and later after divorce.

But you are wise to start negotiating a separation agreement now, since it can provide certainty as to the mutual obligations you will have towards each other once you do formally separate.