Is my state retirement check from WA State (NO State Income Tax) subject to STATE income tax if I live mostly (8+ months.per year) in another state that has a State Income Tax?

The taxing policy of the state from which the check originates has no bearing on the issue, except that the other state may consider you retirement plan as qualified or not qualified depending on the type of retirement plan. The policy of the other state governs. Some states in my area at least, have a clearly defined process and formula for computing tax due for part-time residents.

As SmearCase says, the state where you LIVE the bulk of the time will probably have a set policy as to how to tax such a pension.