Although there are easier ways to solve this equation in MATLAB (e.g., using dsolve()), you mentioned Laplace transform in your question. The following code shows how to use it Laplace, and inverse Laplace transform to find the impulse response.
syms y(t) f(t) Ly
dy = diff(y, 1);
ddy = diff(y, 2);
ic = [0 0]; % initial condition: y(0)=0, y'(0)=0
eq = ddy + dy + y == f; % differential equation
eq_impulse = subs(eq, f(t), dirac(t)); % substitude f(t)=dirac(t) i.e., impulse input
eq_laplace = laplace(eq_impulse); % laplace transform of equation
eq_laplace = subs(eq_laplace, [y(0) dy(0) laplace(y)], [ic Ly]); % substituting initial conditions
Ly = solve(eq_laplace, Ly); % seperating the laplace(y) term
y(t) = ilaplace(Ly); % inverse laplace transform
fplot(y, [0 10]); % plot from t=0 to t=10
