1) What is the best linux software to write UI?
a) Is Java suggestable???
b) Does linux QT help me in this regard???
If your application is in C++, then it would make sense to use a portable GUI tool that is also written in C++, like Qt. Why create additional headaches by using one language for the computation and a different one for the GUI?

2) What is the best linux software to plot graphs??
Gnuplot is good if you want a stand-alone program. If you want a Qt-based C++ library that you can use to embed graphs into your program, look at QCustomPlot, Qwt, or Qt's commercial QtCharts libraries.