The query below in VBA should return a string that looks like "Q2 1990" for each record. All credits go to @EvilGenius for this one.

"Q" & DatePart ( "q", [MyDate] ) & " " & DatePart ( "yyyy", [MyDate] )
Just make sure your quarter field "QTR_YR" is a Text/String type. If the format "Q2 1990" is not to your taste, you can tweak the format. Concatenation in VBA is done by the ampersand "&" not "+" like most other languages.
Let us know if this works.
Update:
As requested, the solution in ArcPy with the UpdateCursor would be:
import arcpy
import datetime
def GetQuarter(month):
"""
Define a function to return the quarter of a month (Jan=1, Dec=12).
Where:
0-3 = Q1
4-6 = Q2
7-9 = Q3
10-12 = Q4
"""
if 0 <= int(month) <= 3:
return 1
elif 4 <= int(month) <= 6:
return 2
elif 7 <= int(month) <= 9:
return 3
elif 10 <= int(month) <= 12:
return 4
else:
raise ValueError, "Month needs to be between 0-12"
fc = r"C:\Users\MyUsername\Desktop\Working.gdb\TestPoints" # Change to path to your feature class.
with arcpy.da.UpdateCursor(fc, ["MyDate", "MyQuarter"]) as cursor: # Change "MyDate" and "MyQuarter" to your respective field names.
for row in cursor:
date = row[0] # A datetime object
quarter = "Q{0} {1}".format(GetQuarter(date.month), date.year) # The quarter string will look like "Q2 1990".
row[1] = quarter # Assign quarter value to current row.
cursor.updateRow(row) # Update (save) row.
del cursor