External TCP/IP with GSM modem GPRS connection

In our project currently ,we are using GSM modems GPRS connection for upload data to server using HTTP protocol(GSM has internal HTTP).
But now are planning to move to MQTT protocol for upload data to server.For that we are using external TCP/IP protocol on micro controller.

Here my doubt is How to use GSM modems GPRS connection to External TCP/IP protocol on microcontroller. Our SIM has Private APN with static IP address . I tried connect using this static IP address but always TCP connection failure.

Note:Our GSM Modem connected with controller through UART connection.

Please Anyone help me to solve this problem...

Thanks in advance...

We dont know what GSM modem or GPRS stuffs you are using. Give us some link.

Actually I am using u-blox SARA-g350 (2G) module.