is there a deciduous forest near Seattle?

Alder trees are the first to inhabit an area burned by fire, or logged off*; but it would be hard to find one of these areas because fir trees move in fairly soon after the alders replace the nitrogen in the ground.

*but often these are replanted with firs by man

You could find small patches; or small patches of maples. We have no large stands of oaks, elms, or birch. Cottonwood is being raised commercially for pulp; but I would not call it a forest. {Nor is an apple orchard considered a forest - lol.}

You might stand a chance if you look on the east side on the Cascade mountains.

The most common native species of trees are evergreens so you will seldom find a natural deciduous forest. Probably the best example of a deciduous forest is the native garry oak forest around JB Ft. Lewis-McChord - the only oak native to Washington.

There are mixed coniferous/deciduous forests, but its hard to find pure deciduous forests anywhere on the west coast.