Estimation of quantile regression by hand Let us suppose we have following data
x   y
1   5
2   4
3   5
4   4
5   7

I would like to   do  Quantile regression in excel,  I have found following information about given method

based on  this information let choose $q=75$ or  75% quantile, in excel I have done following  structure, first  create dummy variable  based on indicator  function and also choose  arbitrary values of alpha and beta 
$$\alpha=0.1 $$
$$\beta=0.2 $$
dummy variable has been  filled using following method

initially all values  are zero,  now I have calculated  one column  for sum 

and finally I have calculated  sum of all  those values, and  then using solver I have estimated coefficients which  minimizes sum, I have got following values and result

please tell me  if I am wrong, and also  statistically  could you explain me  please what does this mean? What does those coefficient  describe in terms of  quantile?  
 A: (A little bit more a long comment than an answer, but I'm missing the repetition to comment)
First, your calculation of the loss appears to be correct (this is R code):
y <- c(5, 4, 5, 4, 7)
x <- c(1, 2, 3, 4, 5)
a <- 0.217092
b <- 1.594303
tau <- 0.75
f <- function(par, y, x, tau) {
    sum((tau - (y  <= par[1] + par[2]*x)) * (y - (par[1] + par[2]*x)))  
}
f(par=c(a, b), y=y, x=x, tau=tau)
[1] 3.782908

Second, there seems to be a problem with the Excel solver. Using R's optimizer, we find:
optim(c(0.1, 0.2), f, y=y, x=x, tau=tau)
$par
[1] 4.4999998 0.4999998

$value
[1] 1.250001

$counts
function gradient 
 143       NA 

so the loss is lower using optim than using Excel's solver.
Third, note that your approach of estimating quantile regression is inferior to solving the corresponding linear program. Anyway, a comparison with Roger Koenker's quantregpackage yields:
library(quantreg)
rq(y ~ x, tau=tau)
Call:
rq(formula = y ~ x, tau = tau)

Coefficients:
(Intercept)           x 
        4.5         0.5 

Degrees of freedom: 5 total; 3 residual

which is very close to the solution of R's optim solver.
About your other question: could you elaborate what exactly you want to understand?
A: solving solution using matlab,  on the base of @BayerSe (thanks my friend,i always respect humans who share their knowledge to others), i have solved this problem in matlab


*

*define objective function
f=@(a) sum((q-(y<=a(1)+a(2)x)).(y-a(1)-a(2)*x))
make initial  guess of   $\alpha$ and $\beta$  and $q$
a_b = [0.1,0.2];

$q=0.75$
and finally  solve  
options = optimset('PlotFcns',@optimplotfval);
x = fminsearch(f,a_b,options)

x =

    4.5000    0.5000


