I agree with Andrey. You could compare your sample's amplification plot to your no-template control (NTC; where you add the SybrGreen and primers but not cDNA). In the case of unspecific amplification, the Ct values would be very similar between the sample and NTC. Running on an agarose gel alongside samples with specific amplification (where there are both amplification plot and melting curve) would be good but if you suspect all your samples have unspecific amplification, running the gel with a DNA ladder and determining the size of the band (if any exists) would suffice.
I agree with the need to run your PCR products on a gel. The best would be to have a positive control, a negative control and NTCs. If you get the expected amplification for the positive control but not for your samples, either you have inhibitors in the DNA extract or you do not have the target DNA.
Then, you need to compare the profil obtained for each samples with those of the NTCs. Do you get also a nice amplification curve without a Tmelting for the NTCs? I would say that you are getting primer dimer.
To discard the possibility of having ihnibitors in the samples DNA extract, do a dilution series of your DNA and try again the PCR. You can also spike an aliquot of the sample extract DNA with a 1/10 vol of the positive control. If no amplification will be detected, than inhibitors are present and you need to clean the DNA.