Expand Matrix table all level in hierarchy - Always Visible while changing Column using Parameter

This can be achieved but you need to construct a helper table that puts all your years and quarters into a single column and then performs a calculate depending on type (quarter vs. year). This allows a filter to just operate on a single field, which won't reset matrix visuals (unlike a parameter, which is switching out fields, thus altering the matrix structure and causing a reset).

Here are steps I took to achieve the above.

Table

Dates (mark as date table)

Dates = 
GENERATE(
    CALENDAR( DATE( YEAR( MIN( 'Table'[Date] ) ), 1, 1 ), DATE( YEAR( MAX( 'Table'[Date] ) ), 12, 31 ) ),
    VAR _dt = [Date] RETURN
    ROW(
        "Year", YEAR( _dt ),
        "YearTxt", FORMAT( _dt, "YYYY" ),
        "QuarterNum", QUARTER( _dt ),
        "Quarter", "Q" & QUARTER( _dt )
    )
)

Output:

DateHierarchySelect (helper table)

DateHierarchySelect = 
VAR _yrs = 
    GENERATE( 
        //using selectcolumns so column name is Period instead of YearTxt
        SELECTCOLUMNS( DISTINCT( Dates[YearTxt] ), "Period", Dates[YearTxt] ) , 
        ROW( "Period Scale", "Year" ) 
    )
VAR _qts = 
    GENERATE( 
        DISTINCT( Dates[Quarter] ), 
        ROW( "Period Scale", "Quarter" ) 
    )
RETURN
UNION( _yrs, _qts )

Output:

Relationships

Create a measure that iterates through DateHierarchySelect[Period Scale] and provides the scale-specific calc. In this instance we are just adding Sales. You may need a different iterator and different calculation depending on your objectives, but I think this provides the gist.

Sales_DynamicPeriod = 
SUMX( 
    VALUES( DateHierarchySelect[Period Scale] ), 
    SWITCH( 
        DateHierarchySelect[Period Scale], 
        "Year", 
            CALCULATE( 
                SUM( 'Table'[Sales] ), 
                TREATAS( VALUES( DateHierarchySelect[Period] ), Dates[YearTxt] ) 
            ),
        "Quarter", 
            CALCULATE( 
                SUM( 'Table'[Sales] ), 
                TREATAS( VALUES( DateHierarchySelect[Period] ), Dates[Quarter] ) 
            )
    )
)

Now, put together your slicer on DateHierarchySelect[Period Scale] and matrix with all levels expanded.