For an infinitesimal transformation in phase space, what functions are allowed for this to be a canonical transformation? Consider an infinitesimal transformation:
$$(q_{i},p_{j}) \quad\longrightarrow \quad(Q_{i},P_{j}) ~=~ \left(q_{i}  + \alpha F_{i}(q,p),~p_{j} + \alpha E_{j}(q,p)\right) $$
where $α$ is considered to be infinitesimally small.
Now, if we construct Jacobian matrix, we will have:
$$
\jmath  =\begin{pmatrix}
\delta_{ij}+ \alpha{\frac{\partial F_{i} }{\partial q_{j}}} &  \alpha{\frac{\partial F_{i} }{\partial p_{j}}} \\ \alpha{\frac{\partial E_{i} }{\partial q_{j}}}
 & \delta_{ij}+ \alpha{\frac{\partial E_{i} }{\partial p_{j}}}
\end{pmatrix}.$$
What functions $F_{i} (q, p)$ and $E_{i} (q, p)$
are allowed for this to be a canonical transformation?
To be canonical transformation, it's required to hold:
$$\jmath j \jmath^{T} = j$$
in which $  j = \begin{pmatrix}
0 & 1\\ 
 -1&0 
\end{pmatrix}$.
To hold the canonical transformation, there should be:
$$\frac {\partial F_{i}}{\partial q_{j}} = - \frac {\partial E_{i}}{\partial p_{j}} $$
which is true if
$$F_{i} = \frac {\partial G}{\partial p_{i}}  \; \; , \; \; E_{i} = - \frac {\partial G}{\partial q_{i}}  $$
for some function $G(q, p)$. 
Now my problem is that by calculating everything I can't figure out how to reach to last two formulas. The formulas which shows the possibilities for $F_{i}$ and $E_{i}$?
 A: *

*First of all, be aware that there exist various different definitions of canonical transformations (CT) in the literature, cf. e.g. this Phys.SE post. What OP (v3) above refers to as a CT, we will in this answer call a symplectomorphism for clarity. What we in this answer will refer to as a CT, will just be a CT of type 2.


*It is possible to show (see e.g. Ref. 1) that an arbitrary time-dependent infinitesimal canonical transformation (ICT) of type 2 with generator $G=G(z,t)$ can be identified with a Hamiltonian vector field (HVF)
$$  \delta z^I~=~\varepsilon\{ z^I,G\}_{PB}~\equiv~ \sum_{K=1}^{2n} J^{IK} \frac{\partial G}{\partial z^K} , $$
$$ X_{-G}~\equiv~-\{G,\cdot\}_{PB}~\equiv~\{\cdot,G\}_{PB},\tag{1} $$
with (minus) the same generator $G$. Here $z^1,\ldots, z^{2n}$, are phase space variables, $t$ is time, $\varepsilon$ is an infinitesimal parameter, and $J$ is the symplectic unit matrix, $$\tag{2} J^2~=~-{\bf 1}_{2n\times 2n}.$$


*A general time-dependent infinitesimal transformation (IT) of phase space can without loss of generality be assumed to be of the form
$$ \tag{3} \delta z^I~=~\varepsilon \sum_{K=1}^{2n} J^{IK} G_K(z,t) ,\qquad I~\in~\{1,\ldots, 2n\}, $$
because the matrix $J$ is invertible.


*Next consider a time-dependent infinitesimal symplectomorphism (IS), which can be identified with a symplectic vector field (SVF). It is possible to show that a SVF [written in the form (3)] satisfies the Maxwell relations$^1$
$$\tag{4} \frac{\partial G_I(z,t)}{\partial z^J}~=~(I \leftrightarrow J),\qquad I,J~\in~\{1,\ldots, 2n\}.  $$


*Eq. (4) states that the one-form
$$\tag{5}  \mathbb{G}~:=~ \sum_{I=1}^{2n}G_I(z,t) \mathrm{d}z^I$$
is closed
$$\tag{6} \mathrm{d}\mathbb{G}~=~0. $$


*It follows from Poincare Lemma, that locally there exists a function $G$ such that $ \mathbb{G}$ is locally exact
$$\tag{7}  \mathbb{G}~=~\mathrm{d}G. $$
Or in components,
$$\tag{8}  G_I(z,t)~=~\frac{\partial G(z,t)}{\partial z^I},\qquad I~\in~\{1,\ldots, 2n\}  .$$


*In summary we have the following very useful theorem for a general time-dependent infinitesimal transformation (IT).

Theorem. An infinitesimal canonical transformation (ICT) of type 2 is an infinitesimal symplectomorphism (IS). Conversely, an IS is locally a ICT of type 2.



*2D counterexample: Consider the phase space $M=\mathbb{R}^2\backslash\{(0,0)\}$ with the symplectic 2-form $\omega =\mathrm{d}p\wedge \mathrm{d}q$. One may check that the vector field
$$X=\frac{q}{q^2+p^2}\frac{\partial}{\partial q} +\frac{p}{q^2+p^2}\frac{\partial}{\partial p} $$
is SVF/IS but it is not a HVF/ICT of type 2. The problem is that the candidate ${\rm arg}(q+ip)$ for the Hamiltonian generator is multi-valued, and hence not globally well-defined.
References:

*

*H. Goldstein, Classical Mechanics; 2nd eds., 1980, Section 9.3; or 3rd eds., 2001, Section 9.4.

--
$^1$ OP already listed some (but not all) of the Maxwell relations (4) in his second-last equation. All of the Maxwell relations (4) are necessary in order to deduce the local existence of the generating function $G$.
