According to some definitions of periodic motion on internet as well as in my book:
Motion repeated in equal interval of time is called Periodic Motion.
Now, if I am in uniform linear motion my velocity is constant i.e. my displacement (w.r.t origin) changes with same rate. If I move 10m in first second then I will move 10m in next second also.
So isn't this a periodic motion ?
Periodic motion means that if you are at position $x$ at time $t$, you will be at position $x$ again at time $t+T$, where $T$ is the period.
So no.
Unless you want to stretch things and consider $T = \infty$. But if you do that, all motion is periodic. Still, it is a limiting case that comes up in Fourier Analysis. For a good introduction, see But what is the Fourier Transform? A visual introduction. by 3Blue1Brown.
Another limiting case is if $v = 0$. This case also comes up in Fourier Analysis.
Periodic motion means that if the state of the system is $\Upsilon(t)$ at some time $t\in\mathbb{R}$ then $\exists T\in\mathbb{R}_+\cup\{ 0\}$ such that $\Upsilon(t+nT)=\Upsilon(t)$, $\forall n\in\mathbb{Z}.$ This means that the system has to come back to the same state that it is in at any given time after every interval of $T$, called the period of the periodic motion.
The state of the system, informally speaking, is the information that describes all there is to know about the system. For deterministic physics, this means that the state of the system is what you need to know about the system to be able to predict its future/past. You can see a more detailed discussion on the concept of the state of a system in physics on this post.
For a classical system, the state of the system comprises of its generalized coordinates and generalized velocities $(q_i,\dot{q}_i)$ or, equivalently, its canonical coordinates and canonical momenta $(q_i,p_i)$. So, for a classical system of a single particle, a periodic motion means that both its position and velocity (or, equivalently, its position and momentum) have to repeat the values that they have at any given time after each time-interval of some period $T$, i.e., $\exists T\in\mathbb{R}_{+}\cup\{0\}$ such that $q(t)=q(t+nT)$ and $p(t)=p(t+nT)$ has to hold true $\forall t\in\mathbb{R},\forall n\in\mathbb{Z}$ where $q(t),p(t)$ are the position and the momentum of the particle respectively.
As you can see, this criterion is clearly not satisfied by a particle in a uniform motion moving in ordinary $\mathbb{R}^3$ space. However, if the topology of the space in which it is moving has non-trivial global features then such a motion can be periodic. For example, a particle moving uniformly in an $S^1$ space (i.e., a circle) would be performing a periodic motion.
An object or system displaying periodic motion needs to have a period (hence the term periodic).
So the body must repeat its motion after each period, or the body must come back to a certain point along a path after each period. So an object moving with the same velocity away from an origin indefinitely does not qualify as periodic motion. Indeed you could have chosen any arbitrary distance and associated a time (also arbitrary) with it, as oppose to periodic motion where the period is a characteristic of the motion.
A good example of periodic motion is a simple pendulum. The bob oscillates about a fixed point and returns after a fixed time.
