How does two convolution layers make computation quadratic increase in GoogLeNet?
For any CNN layer $n$ with $F_n$ filters and kernel size $K_n^{height},K_n^{width}$ the number of parameters to train is 
$$F_{n-1} \cdot F_n \cdot K_n^{height} \cdot K_n^{width} + F_n$$

where the first summand stands for the weights and the second for the bias parameters (if $n$ is the first CNN layer then $F_{n-1}$ equals the number of input channels). And for the special case of 


$F_{n-1} = F_n$ (i.e. same number of filters for layer $n$ and layer $n-1$)
and $K_n^{height} = K_n^{width}$ (i.e. square shaped Kernel)


the number of parameters for layer $n$ is $F_n^2 \cdot K_n^2 + F_n$. Accordingly, you have a non-linear dependency of trainable parameters with regards to the number of filters.