Do x-rays get emitted upon initial acceleration in this setup? Based on the diagram and the paragraph before it, the x-rays are caused by decelerating the moving electrons and thus x-rays are emitted where the diagram clearly shows it (the red arrows pointing downwards labeled "x-rays"). But does this mean that x-rays are also emitted near the cathode where the electrons are initially accelerated?

 A: 
As with any vacuum tube, there is a cathode, which emits electrons into the vacuum and an anode to collect the electrons, thus establishing a flow of electrical current, known as the beam, through the tube. A high voltage power source, for example 30 to 150 kilovolts (kV), is connected across cathode and anode to accelerate the electrons. The X-ray spectrum depends on the anode material and the accelerating voltage

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Electrons from the cathode collide with the anode material, usually tungsten, molybdenum or copper, and accelerate other electrons, ions and nuclei within the anode material. About 1% of the energy generated is emitted/radiated, usually perpendicular to the path of the electron beam, as X-rays. The rest of the energy is released as heat. 

(Italics mine.)
Radiation needs  acceleration or deceleration .
You ask:

But does this mean that x-rays are also emitted near the cathode where the electrons are initially accelerated?

The beam leaving the cathode and undergoing continuous acceleration to the anode, will radiate as it is accelerated, but to a much smaller amount of energy loss.( See this link,the discussion for linear accelerators.) The amount is small and not in the x-ray electromagnetic spectrum, rather the infrared, (as the beam does not shine contiunuously) for a cathode ray tube.  So no xrays   either beginning at the cathode or on the electron beam line.
The reason x-rays are produced at the anode is that for a percentage of the hits (1% quoted above), the sudden deceleration on hitting the anode takes up (for that percentage) most of  the energy of the beam electron, and thus radiates in the keV X-ray region, (due to the keV voltage drop between cathode and anode) when interacting with the charges in the anode.
This is a plot of the spectrum of photons generated given and initial KV value:

As a sidenote, the small radiation effects in linear accelerations in contrast to circular ones makes linear colliders attractive because there is little energy loss in the beams, in contrast to the synchrotron radiation of circular ones. 
A: The X-rays are emitted as a form of energy release. In decelerating, the electrons "lose" kinetic energy, and we see this energy released in the form of X-rays. To accelerate the electrons would require an input of energy. We do not expect X-rays to be released in this case (an output of energy).
A: The acceleration of the electrons whilst travelling between cathode and anode is very much less than the (negative) acceleration of the electrons when they hit the target atoms.
The higher the acceleration the more energetic are the emitted photons.
So the photons emitted from the interaction of the electrons with the target are much more energetic (in the X-ray region) than the photons emitted when the electrons are traversing the gap between the cathode and the anode.  
The text hints at this when it states "X-rays are produced when the electrons are suddenly stopped in the anode" after mentioning the acceleration of the electrons between cathode and anode.
