The following list details which Q plates where issued during each year.
Q plates where issued from 1921 to 1955.
1921 – Q1 to Q8182
1922 – Q8183 to Q13922
1923 – Q13923 to Q21434
1924 – Q21435 to Q23724
1925 – Q32725 to Q47920
1926 – Q47920 to Q67735
1927 – Q67736 to Q86509
1928 – Q86510 to Q104715
1929 – Q104716 to Q124795
1930 – Q124796 to Q139536
1931 – Q139537 to Q151572
1932 – Q151573 to Q163578
1933 – Q163579 to Q177046
1934 – Q177047 to Q195314
1935 – Q195315 to Q216214
1936 – Q216215 to Q237582
1937 – Q237583 to Q260534
1938 – Q260535 to Q287228
1939 – Q287229 to Q312857
1940 – Q312856 to Q333259
1941 – Q333260 to Q346728
1942 – Q346729 to Q361620
1943 – Q361621 to Q378113
1944 – Q379114 to Q390353
1945 – Q390354 to Q403308
1946 – Q403309 to Q424985
1947 – Q424986 to Q442331
1948 – Q442332 to Q460951
1949 – Q460952 to Q483074
1950 – Q483075 to Q521342
1951 – Q521343 to Q557851
1952 – Q557852 to Q588327
1953 – Q588328 to Q625196
1954 – Q625197 to Q669281
1955 – Q669282 to Q687612
Additional information
According to the Wikipedia entry, all Q plates from 1921 to 1924 were issued on a black base with white characters. It was only in 1924 (or 1 April 1925 according to this PPQ publication) that they changed to the more familiar Q plate design with the black Q on a white background, followed by white digits on the back background.
There are also numerous reports that at some stage the Q was painted red. In this case the plates either had white digits on a black background or black digits on a white background.
The first plate, Q1, was issued 1 July 1921. The final plate, Q687612, was issued 31 May 1955.