As you have already discovered packing a chromatographic column is not an easy task. Therefore the concept of cleaning in place has been developed. Here the packed column materials are cleaned for fouling proteins and other contaminants without unpacking the column. Each type of columns has different chemical stabilities and therefore needs to be treated differently. The column supplier has provided detailed description for each type of columns.
For the Q Sepharose fast flow (FF) materials you are using the link is: Q sephrose FF description and CIP protocol..
Remember to consider the risk of injecting strong acid or base into a pressured system. What precautions are you taking?