Hi,
thank you for providing parts of your results. The q-value is defined as the "minimal False Discovery Rate at which a peptide with a given score would be accepted". It is a "local" property of a specific peptide not a property of the "global" full set of results, which usually makes it a better measure of confidence for a given peptide. That means, if a peptide (or PSM) has a q-value of 0.05, it first appears in the set of accepted peptides, when you set the FDR threshold to 0.05 (or higher). By default our FDR tool calculates this q-value and if you use IDFilter to filter the peptides, it will usually use the last score calculated for a peptide (unless specified otherwise). That is why your results have all q-values lower than 0.01 (with such a filter).
In case you are not familiar with the concept of a False Discovery Rate:
In our pipelines we usually use the target-decoy concept to simulate false discoveries. You include decoys in the database that you know are not present (e.g. reversed peptides from your original "target" database) and use the amount of identified decoy peptides as a proxy for the amount of falsely identified target proteins. The FDR then is only a measure of the proportion of falsely identified peptides in your overall accepted peptides (i.e. usually peptides above a threshold).
A small tutorial can be found here: http://www.inf.fu-berlin.de/lehre/WS14/ProteomicsWS14/LUS/lu7c/433/index.html
Cheers
Julianus