I have a recipe that specifically calls for a 4-quart slow cooker, however the slow cooker that I own is a 6-quart. Will using a 6-quart slow cooker make a difference, or does it not matter?
Size does matter in a slow cooker. In order for it to cook effectively a slow cooker has to be at least 2/3 full, other wise it cooks hot and will generally dry out faster. If you are cooking at less than 2/3 full you need to adjust your heat settings and/or depending on the recipe adjust the amount of liquid. Cooking meats becomes especially troublesome when underfilling because they tend to dry out and end up a lot tougher than expected. If you are cooking stews or chilis then simply adding more liquid to your recipe will generally work.
Unless you are using recipes at the level of those by Adria, Blumenthal, or Achatz--with their attendant need for absolute precision--you should be absolutely fine. You may wish to slightly adjust the amount of liquid included to allow for slightly greater water evaporation from the larger surface.
Alternatively, provide the recipe (just edit your question) so we can check more thoroughly.