We present seven new transit light curves of the hot-Jupiter TrES-5b observed with the 2-m Himalayan Chandra Telescope, Hanle, India during 2016-2020 to look for the transit timing variation (TTV). For this study, total 59 transit light curves have been considered which include seven light curves from our new observations, 37 from the literature, and 15 from the Exoplanet Transit Database (ETD). In order to have homogeneously derived mid-transit times for precise TTV study, all the light curves were analyzed with a uniform procedure using the Transit Analysis Package (TAP). By fitting a linear ephemeris model to the mid-transit time data, we derive new transit ephemeris with $\chi^{2}_{red}= 2.43$ for TrES-5b, which is consistent and even more precise than the previous results. The best-fit ephemeris with $\chi^{2}_{red} >1$ indicates that the linear ephemeris model does not represent the transit time data well. To look for the periodicity in the timing residuals of linear ephemeris model, a generalized Lomb-Scargle periodogram is computed. However, we have not found any significant periodicity in the timing residuals. The highest peak power obtained in the periodogram at the frequency of 0.005232 rad/period has the false alarm probability (FAP) of 12%, which is found below from the threshold values (i.e. FAP =5% and 1%). This result enable us to conclude that the additional planet might not be present in the TrES-5 system. To confirm this, the further high-precision follow-up observation of the TrES-5 system would be required.