Depends on your definition of social justice. There are those for whom it means everyone gets equal results. For them, the answer to your question is unequivocally yes. For those with more flexible definitions the answer is probably no.
I don't think they are conflicting, rather nature's conservation is for ocial equality. Natural ecosystems are useful to people for many reasons. They provide water, food, building materials, energy and medicine. They regulate the climate. They also have important cultural significance. All these reasons give nature an economic value for humans. Even when this value is not apparent, because most and key services of nature are not traded on markets, nature's worth should be taken into consideration in dealing with nature. It has long been known that the higher the average income is within a society, the greater the economic value nature has for that society. That is because in valuing nature's benefits in economic terms, the benefits of what nature provides are compared with the benefits provided by consumption goods. Those who have higher incomes can consume more and will therefore normally ascribe a higher value to nature.
Now if the inequality of income distribution influences the economic value of nature, then it must be noted that if the services provided by nature for human well-being can be substituted for well with human-produced goods and services, then the economic value of nature for a society is higher, the more equally incomes are distributed within it. The reverse is also true. The more unequally income is distributed in a society, the less the economic value that society will place on nature.
So there is a clear relationship between social equity and nature conservation. The income inequality will lead nature to be undervalued. By reducing income inequality, the valuation of nature in economic terms will consequentially rise, resulting in greater value being placed on nature in decisions about measures to promote economic development.