I applied the real-time product on 2.5% agarose gel and only targeted the specific band, but even with changes in primer concentration, cDNA, and temperature gradient, these shoulder are still present.
Is the shoulder problematic? That's the real question.
It's at ~65, so it's primer dimers. It's also not huge, so it's "not a lot of primer dimers".
The only issue is if the primer dimers amplify alongside your amplicon, or only later when most other reagents are exhausted. If it's the latter, then it doesn't matter.
I would run a serial dilution of your cDNA, starting with maybe a 1/5 of neat stock, and then serially diluted by 2-fold (or 10-fold, if your target is abundant enough). If you get Cq values in line with expected dilution, then you can safely ignore the primer dimers.
I agree with John. It could help to increase the annealing temperature. Additionally, I would stop the PCR just before the start of the plateau phase and make a melt curve analysis then. This will show you if the unspecific products (presumably primer dimers) are produced early or late in the PCR. If they occur late, then it's not at all a problem for quantification. But even if they appear early, the faint amount is very unlikely problematic.