An $\alpha$-particle is a helium nucleus, represented as $^{4}_{2}He$. A proton is represented as $^{1}_{1}H$ or $^{1}_{1}p$. The nuclear transmutation reaction can be written as:

$^{14}_{7}N + ^{4}_{2}He \rightarrow ^{A}_{Z}X + ^{1}_{1}H$

According to the law of conservation of mass number and atomic number in a nuclear reaction: Conservation of mass number (superscripts): $14 + 4 = A + 1 \implies A = 17$ Conservation of atomic number (subscripts): $7 + 2 = Z + 1 \implies Z = 8$

The element with atomic number $Z = 8$ is Oxygen ($O$). Therefore, the newly formed atom $^{A}_{Z}X$ is $^{17}_{8}O$.