At this wave numbers you will also see water on silica. I would try to look for the N-H and C-H bands (~3300 and ~2970 cm-1), but I guess they will also be difficult to see. If your surface area and loadings are low I really wouldn't expect to see anything by FTIR. Zeta potential is a lot better. Compare your pre and post treated material, potential should go from negative to positive at neutral pH. Another way is to do a chemical test: Kaiser or ninhydrin test (typically used in peptide synthesis to check for Fmoc deprotection): if you have amines your solid will turn from white to dark blue.
You have used always highly basic conditions to coat silica nanoparticle with a shell silica.. Did you washed the resulting materials, After the reaction of the silica, with APTES , the surface potential cannot be positively charged excepted if the material have been washed with a solution of HCl 5% 5mL in mL EtOH..
Now you can passivate the residual silanol groups of the resulting 3amino-propyl functionalized material surface by washing it with a mixture of EtOH/water , then with EtOH , then with ethylether, then after drying at 60°C , dehydration under vacuum at 120°C , then reaction with Chlorotrimethylsilane... then washing accordiing the previous protocole....without washing with HCl /EtOH ( the HCl formed during the trimethylsilylation leading toi teh protonated amino groups ( ie Ammonium ones...)
As I remembered, the absorption peak at 1600 cm-1 represents the amide groups (CO-NH((1or2)), but the groups on APTES are amine groups (CH2-NH2). This is why you did not see any difference before and after functionalization. Like Igor mentioned, you may need to have a look at your FTIR spectra again to see if you can see the peak from CH2 at 2900 cm-1. The CH2 groups are introduced by the carbon chain from APTES. These peaks normally turns out as a group of 2 small peaks.
Zeta potential is more straightforward because your aim is to get the positively charged silica particles.
Another simple measurement is the TGA. From the weight loss between non-functionalized silica and amine functionalized silica, you can actually calculate the amount of amine groups grafting on the surface of silica particles. This is also the way to show the success of the modification.
After the measurement on zeta potential and TGA, you may find the problem. If the silica particles are not positively charged, then the grafting rate of APTES is low and the amount of amine groups calculated from TGA should be low. You may have problems with the synthesis process. If you get positively charged silica particles, then you need to find a correct way to characterize your samples and the results from TGA will tell you how many amine groups on the surface can achieve positively charged silica particles.