Can you declare a constant array in VBScript?

An array is the result of a function call (Array()) in VBScript. Only literal values can be made Const. So: No, you can't.

Why not just declare the array as public and then assign the array during the start of the script?

Public arrQuarters
arrQuarters = Array("Q1", "Q2", "Q3", "Q4")

For Each Quarter in arrQuarters
    wscript.echo Quarter
Next

You could define a function to return the array you want to use as a constant. For example:

For Each q in AllQuarters
    wscript.echo q
Next

wscript.echo "element 0 = " & AllQuarters()(0)

AllQuarters()(0) = "X1"

wscript.echo "element 0 still = " & AllQuarters()(0)


Function AllQuarters()
    AllQuarters = Array("Q1","Q2","Q3","Q4")
End Function