No, you can't use in-combat saving throws to respond to disease exposure.
In combat, the saving throws you're granted specifically are only used on effects that have a "save ends" condition (or "save ends both/all").(1) Those are the saving throws you're granted at end of turn, and the saving throws granted by other effects.
Disease exposure isn't an effect with a "save ends", so your in-combat saving throws don't get to be used on it. Instead, it's just an exceptional circumstance that says to roll a single saving throw at the end of combat — and then you get the disease, or you don't.
Even if you could do this, it's almost always going to be a bad idea, since you've got very little to gain from rushing it. Disease exposure doesn't do anything in and of itself.(2) Sooner or later, you'll make a check and be diseased or not. But until you do, you're guaranteed to be 100% disease-free and totally fit for combat. You could rush to that disease check sooner, and maybe you'd get some peace of mind that you won't be diseased — but you've got a slightly below 50% chance of immediately contracting a disease mid-combat. That's probably going to be pretty bad for you, and you'd almost always be better off waiting until after combat to risk contracting a disease.
(1): from RC p218, "Saving throws", the "When" section: ‘An effect that a save can end includes one of the following notations: "save ends," "save ends both," or "save ends all." (later in the same section...) Sometimes a power or other game feature allows a character to make a saving throw immediately against an effect that a save can end.’
(2): I'm not aware of any monsters that care if you're merely exposed, but my monster knowledge in 4e is not encyclopedic.