I'm far from expert in finance, so please bear with me.
It's my understanding that a company who issued stock typically support it when the stock price go down.
Does the same thing happen with government bonds? Or how does it work?
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Companies do not support their stock. Once the security is out on the wild (market), its price fluctuates according to what investors think they are worth.
Support is a whole different concept, financially speaking:
Support or support level refers to the price level below which, historically, a stock has had difficulty falling. It is the level at which buyers tend to enter the stock.
So it is the lowest assumed price for that stock. Once it reaches its price, buyers will rush to the stock, raising its price. The company wants to keep the stock price at acceptable levels, as it can be seen as the general view of the company's health. Also several employees/executives in the company have stock or stock options, so it is in their interest to keep their stock price up.
A bond that goes down in value may indicate a believe the bond issuer (government in this case) won't honor the bond when it matures.
As for bonds, there is a wealth of reading in this site:
who issued stock typically support it when the stock price go down.
No, not many company do that as it is uneconomical for them to do so. Money used up in buying back equity is a wasteful use of a firm's capital, unless it is doing a buyback to return money to shareholders.
Does the same thing happen with government bonds?
Not necessarily again here. Bond trading is very different from equities trading. There are conditions specified in the offer document on when an issuer can recall bonds(to jack up the price of an oversold bond), even government bonds have them.
The actions of the government has a bigger ripple effect as compared to a firm. The government can start buying back bonds to increase it's price, but it will stoke inflation because of the increase in the supply of money in the market, which may or mayn't be desirable. Then again people holding the bond would have to incentivized to sell the bond. Even during the Greek fiasco, the Greek government wasn't buying Greek bonds as it had no capital to buy. Printing more euros wasn't an option as no assets to back the newly printed money and the ECB would have stopped them from being accepted.
And generally buying back isn't useful, because they have to return the principal(which might run into billions, invested in long term projects by the government and cannot be liquidated immediately) while servicing a bond is cheaper and investing the proceeds from the bond sale is more useful while being invested in long term projects. The government can just roll over the bonds with a new issue and refrain from returning the capital till it is in a position to do so.
Without getting to hung-up on terminology here, the management of a company will often attempt to keep stock prices high because of a number of reasons:
Ideally companies keep prices up through performance. In some cases, you'll see companies do other things spending cash and/or issuing bonds to continue to pay dividends (e.g. IBM), or spending cash and/or issuing bonds to pay for stock buybacks (e.g. IBM). These methods can work for a time but are not sustainable and will often be seen as acts of desperation. Companies that have a solid plan for growth will typically not do much of anything to directly change stock prices.
Bonds are a bit different because they have a fairly straight-forward valuation model based on the fact that they pay out a fixed amount per month. The two main reason prices in bonds go down are:
The key here is that bonds pay out the same thing per month regardless of their price or the price of other bonds available. Most stocks do not pay any dividend and for much of those that do, the main factor as to whether you make or lose money on them is the stock price.
The price of bonds does matter to governments, however. Let's say a country successfully issued some 10 year bonds last year at the price of 1000. They pay 1% per month (to keep the math simple.) Every month, they pay out $10 per bond. Then some (stupid) politicians start threatening to default on bond payments. The bond market freaks and people start trying to unload these bonds as fast as they can. The going price drops to $500. Next month, the payments are the same. The coupon rate on the bonds has not changed at all. I'm oversimplifying here but this is the core of how bond prices work.
You might be tempted to think that doesn't matter to the country but it does. Now, this same country wants to issue some more bonds. It wants to get that 1% rate again but it can't. Why would anyone pay $1000 for a 1% (per month) bond when they can get the exact same bond with (basically) the same risks for $500? Instead they have to offer a 2% (per month) rate in order to match the market price.
A government (or company) could in fact put money into the bond market to bolster the price of it's bonds (i.e. keep the rates down.) The problem is that if you are issuing bonds, it's generally (caveats apply) because you need cash that you don't have so what money are you going to use to buy these bonds? Or in other words, it doesn't make sense to issue bonds and then simply plow the cash gained from that issuance back into the same bonds you are issuing. The options here are a bit more limited. I have to mention though that the US government (via a quasi-governmental entity) did actually buy it's own bonds. This policy of Quantitative Easing (QE) was done for more complicated reasons than simply keeping the price of bonds up.