This study proposes a method for estimating and quantifying the uncertainties of a temperature-dependent thermal conductivity. We consider a simple unidimensional problem where a slab is subjected to a constant heat flux applied at both edges. We use a Gaussian process to model the relationship between the thermal conductivity and the temperature. The same Gaussian process acts as a prior, capturing pre-existing knowledge about the thermal conductivity. To address uncertainties, we incorporate a likelihood function that models temperature measurements and their uncertainties using a multivariate normal distribution. The combination of the prior and the likelihood yields a posterior distribution , characterizing the thermal conductivity as a function of the temperature based on observed measurements. We employ the No-U-Turn Sampler (NUTS), a Hamiltonian Monte Carlo (HMC) method, for the solution to the inverse problem. Our results demonstrate the effectiveness of this approach in handling uncertainties, providing an estimation of the thermal conductivity curve and credible intervals for uncertainty quantification.