The stability and stabilization problems for a class of discrete time Markov jump systems (MJSs) are concerned. A general scenario is taken into consideration: the elements of transition probability matrix are partly unknown and signal quantization exists in both control input channel and measurement output channel. The quantized signals in the above mentioned channels are quantized via two different logarithmic quantizers. By virtue of switched Lyapunov function approach, a system mode dependent and quantization error dependent Lyapunov function is constructed to solve the issues of stability analysis and controller design. A set of mode dependent controllers is designed, which is effective in tackling the quantization errors in two channels, and under the circumstance of insufficient information of transition probabilities, the resulting closed loop MJSs are stochastically stable. Finally, a numerical example is utilized to demonstrate the effectiveness of the proposed control strategy.