Could you possibly give an example of how I would go about evaluating an expression to Null?

Yeah, use CompoundExpression where the last element is Null:

CompoundExpression[
    data = {1,3,6,5,3,3,6},
    Print["Median: ", Median[data]],
    Null
]

You're talking about using ; for something else, which I don't want to know anything about because it sounds pretty shady, so I used the explicit function call version of CompoundExpression.

Or alternatively, can you tell me which system file assigns the semicolon to be that infix operator shortcut that you described?

This is not a thing. This behavior is compiled into the WL language interpreter for how it responds to an EnterExpressionPacket (the MathLink/WSTP packet type that handles the In[1] read-eval-print loop). In other words, like most of the system it's not available in source form in the product.

Actually, any expression evaluating to Null will result in no Output cell being created.

If you look at FullForm[Hold[expr;]] you'll see this as CompoundExpression[expr, Null]. In other words, semicolon (;) is actually an infix operator for CompoundExpression, and if you don't provide an expression on its right side, a Null is substituted, and voila, no output.

So, to suppress output, evaluate to Null.